Sample Presentation on Factoring Polynomials

By Paul BartholomewK12 International

Academy Applicant(856) 266-2634

After this presentation you should be able to

bull Explain the concept of factoring polynomialsbull Explain in simple terms why you would need to factor a

polynomialbull Factor polynomials that have a greatest common factor

between terms (ex x3 + 3x2 +x)bull Factor polynomials that have a leading coefficient of 1

(ex x2 + 2x +1)

bull Factor polynomials that have a leading coefficient other then 1 (ex 3x2 + 13x +4)

Factoring a polynomial is the same in concept to factoring a monomial

bull For the monomial 6x2y the prime factors are 23xxy

bull For the polynomial x3 + 2x2 + x the prime factors are x and (x2 + x + 1) written as x(x2 + x + 1)

This presentation will show you how to factor some basic polynomials

Why factor polynomials 1) To reduce expressions to their lowest terms like you do with monomials

6x2y = 23 xxy = 3xy 2x 2x

Monomial Example

Polynomial Examplex3 + 2x2 + x = x(x2 + x + 1) = xx2 + x + 1 (x2 + x + 1)

2) Factoring polynomials is key to solving quadratics and other higher order equations

Factoring a polynomial means expressing it as a product of other

polynomials

This can help you simplify expressions and solve higher order

equations

Summary

Factoring Method 1

Factoring polynomials with a Greatest Common Factor (GCF)

IMPORTANT - Always look for a GCF before using any other

factoring method

Steps1) Find the greatest common factor

(GCF)

2) Divide the polynomial by the GCF The quotient (answer from the division) is the other factor

3) Express the polynomial as the product of the quotient and GCF

3cd(2c2 4cd 1)

The answer should look like this

Ex 6c3d 12c2d 2 3cd

Ex 6c3d 12c2d 2 3cd

GCF 3cdSTEP 1

Step2 Divide

(6c3d 12c2d2 3cd) 3cd

2c2 4cd 1

Factor these on your own looking for a GCF

1 6x3 3x2 12x

2 5x2 10x 35

3 16x3y4z 8x2 y2z3 12xy3z 2

3x 2x2 x 4 5 x2 2x 7

4xy2z 4x2 y2 2xz 2 3yz

A note Method 2 is a way of lsquoworking backwardsrsquo from the way one would multiply

two binomials together

NoticeThe first term of the answer is the product of the (first) variable terms

The last term of the answer is the product of the constant (last) terms

The middle term is the sum of the constant (last) terms

Please keep this in mind as we go forwardhellip

Factoring Polynomials ndash Method 2

How do you factor a trinomial with a leading coefficient of 1

Example Factor x2 ndash 13x + 36

You can use a diamond

The factors are (x ndash 9)(x ndash 4)

Write the middle coefficient here

Write the last term here

ndash13

+36

Now find factors that will multiply to the bottom number and add to the top number

ndash9 ndash4

Example Factor x2 ndash 3x ndash 40 ndash3

ndash40ndash8 +5 The factors are (x ndash 8)(x + 5)

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

After this presentation you should be able to

bull Explain the concept of factoring polynomialsbull Explain in simple terms why you would need to factor a

polynomialbull Factor polynomials that have a greatest common factor

between terms (ex x3 + 3x2 +x)bull Factor polynomials that have a leading coefficient of 1

(ex x2 + 2x +1)

bull Factor polynomials that have a leading coefficient other then 1 (ex 3x2 + 13x +4)

Factoring a polynomial is the same in concept to factoring a monomial

bull For the monomial 6x2y the prime factors are 23xxy

bull For the polynomial x3 + 2x2 + x the prime factors are x and (x2 + x + 1) written as x(x2 + x + 1)

This presentation will show you how to factor some basic polynomials

Why factor polynomials 1) To reduce expressions to their lowest terms like you do with monomials

6x2y = 23 xxy = 3xy 2x 2x

Monomial Example

Polynomial Examplex3 + 2x2 + x = x(x2 + x + 1) = xx2 + x + 1 (x2 + x + 1)

2) Factoring polynomials is key to solving quadratics and other higher order equations

Factoring a polynomial means expressing it as a product of other

polynomials

This can help you simplify expressions and solve higher order

equations

Summary

Factoring Method 1

Factoring polynomials with a Greatest Common Factor (GCF)

IMPORTANT - Always look for a GCF before using any other

factoring method

Steps1) Find the greatest common factor

(GCF)

2) Divide the polynomial by the GCF The quotient (answer from the division) is the other factor

3) Express the polynomial as the product of the quotient and GCF

3cd(2c2 4cd 1)

The answer should look like this

Ex 6c3d 12c2d 2 3cd

Ex 6c3d 12c2d 2 3cd

GCF 3cdSTEP 1

Step2 Divide

(6c3d 12c2d2 3cd) 3cd

2c2 4cd 1

Factor these on your own looking for a GCF

1 6x3 3x2 12x

2 5x2 10x 35

3 16x3y4z 8x2 y2z3 12xy3z 2

3x 2x2 x 4 5 x2 2x 7

4xy2z 4x2 y2 2xz 2 3yz

A note Method 2 is a way of lsquoworking backwardsrsquo from the way one would multiply

two binomials together

NoticeThe first term of the answer is the product of the (first) variable terms

The last term of the answer is the product of the constant (last) terms

The middle term is the sum of the constant (last) terms

Please keep this in mind as we go forwardhellip

Factoring Polynomials ndash Method 2

How do you factor a trinomial with a leading coefficient of 1

Example Factor x2 ndash 13x + 36

You can use a diamond

The factors are (x ndash 9)(x ndash 4)

Write the middle coefficient here

Write the last term here

ndash13

+36

Now find factors that will multiply to the bottom number and add to the top number

ndash9 ndash4

Example Factor x2 ndash 3x ndash 40 ndash3

ndash40ndash8 +5 The factors are (x ndash 8)(x + 5)

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Factoring a polynomial is the same in concept to factoring a monomial

bull For the monomial 6x2y the prime factors are 23xxy

bull For the polynomial x3 + 2x2 + x the prime factors are x and (x2 + x + 1) written as x(x2 + x + 1)

This presentation will show you how to factor some basic polynomials

Why factor polynomials 1) To reduce expressions to their lowest terms like you do with monomials

6x2y = 23 xxy = 3xy 2x 2x

Monomial Example

Polynomial Examplex3 + 2x2 + x = x(x2 + x + 1) = xx2 + x + 1 (x2 + x + 1)

2) Factoring polynomials is key to solving quadratics and other higher order equations

Factoring a polynomial means expressing it as a product of other

polynomials

This can help you simplify expressions and solve higher order

equations

Summary

Factoring Method 1

Factoring polynomials with a Greatest Common Factor (GCF)

IMPORTANT - Always look for a GCF before using any other

factoring method

Steps1) Find the greatest common factor

(GCF)

2) Divide the polynomial by the GCF The quotient (answer from the division) is the other factor

3) Express the polynomial as the product of the quotient and GCF

3cd(2c2 4cd 1)

The answer should look like this

Ex 6c3d 12c2d 2 3cd

Ex 6c3d 12c2d 2 3cd

GCF 3cdSTEP 1

Step2 Divide

(6c3d 12c2d2 3cd) 3cd

2c2 4cd 1

Factor these on your own looking for a GCF

1 6x3 3x2 12x

2 5x2 10x 35

3 16x3y4z 8x2 y2z3 12xy3z 2

3x 2x2 x 4 5 x2 2x 7

4xy2z 4x2 y2 2xz 2 3yz

A note Method 2 is a way of lsquoworking backwardsrsquo from the way one would multiply

two binomials together

NoticeThe first term of the answer is the product of the (first) variable terms

The last term of the answer is the product of the constant (last) terms

The middle term is the sum of the constant (last) terms

Please keep this in mind as we go forwardhellip

Factoring Polynomials ndash Method 2

How do you factor a trinomial with a leading coefficient of 1

Example Factor x2 ndash 13x + 36

You can use a diamond

The factors are (x ndash 9)(x ndash 4)

Write the middle coefficient here

Write the last term here

ndash13

+36

Now find factors that will multiply to the bottom number and add to the top number

ndash9 ndash4

Example Factor x2 ndash 3x ndash 40 ndash3

ndash40ndash8 +5 The factors are (x ndash 8)(x + 5)

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Why factor polynomials 1) To reduce expressions to their lowest terms like you do with monomials

6x2y = 23 xxy = 3xy 2x 2x

Monomial Example

Polynomial Examplex3 + 2x2 + x = x(x2 + x + 1) = xx2 + x + 1 (x2 + x + 1)

2) Factoring polynomials is key to solving quadratics and other higher order equations

Factoring a polynomial means expressing it as a product of other

polynomials

This can help you simplify expressions and solve higher order

equations

Summary

Factoring Method 1

Factoring polynomials with a Greatest Common Factor (GCF)

IMPORTANT - Always look for a GCF before using any other

factoring method

Steps1) Find the greatest common factor

(GCF)

2) Divide the polynomial by the GCF The quotient (answer from the division) is the other factor

3) Express the polynomial as the product of the quotient and GCF

3cd(2c2 4cd 1)

The answer should look like this

Ex 6c3d 12c2d 2 3cd

Ex 6c3d 12c2d 2 3cd

GCF 3cdSTEP 1

Step2 Divide

(6c3d 12c2d2 3cd) 3cd

2c2 4cd 1

Factor these on your own looking for a GCF

1 6x3 3x2 12x

2 5x2 10x 35

3 16x3y4z 8x2 y2z3 12xy3z 2

3x 2x2 x 4 5 x2 2x 7

4xy2z 4x2 y2 2xz 2 3yz

A note Method 2 is a way of lsquoworking backwardsrsquo from the way one would multiply

two binomials together

NoticeThe first term of the answer is the product of the (first) variable terms

The last term of the answer is the product of the constant (last) terms

The middle term is the sum of the constant (last) terms

Please keep this in mind as we go forwardhellip

Factoring Polynomials ndash Method 2

How do you factor a trinomial with a leading coefficient of 1

Example Factor x2 ndash 13x + 36

You can use a diamond

The factors are (x ndash 9)(x ndash 4)

Write the middle coefficient here

Write the last term here

ndash13

+36

Now find factors that will multiply to the bottom number and add to the top number

ndash9 ndash4

Example Factor x2 ndash 3x ndash 40 ndash3

ndash40ndash8 +5 The factors are (x ndash 8)(x + 5)

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Factoring a polynomial means expressing it as a product of other

polynomials

This can help you simplify expressions and solve higher order

equations

Summary

Factoring Method 1

Factoring polynomials with a Greatest Common Factor (GCF)

IMPORTANT - Always look for a GCF before using any other

factoring method

Steps1) Find the greatest common factor

(GCF)

2) Divide the polynomial by the GCF The quotient (answer from the division) is the other factor

3) Express the polynomial as the product of the quotient and GCF

3cd(2c2 4cd 1)

The answer should look like this

Ex 6c3d 12c2d 2 3cd

Ex 6c3d 12c2d 2 3cd

GCF 3cdSTEP 1

Step2 Divide

(6c3d 12c2d2 3cd) 3cd

2c2 4cd 1

Factor these on your own looking for a GCF

1 6x3 3x2 12x

2 5x2 10x 35

3 16x3y4z 8x2 y2z3 12xy3z 2

3x 2x2 x 4 5 x2 2x 7

4xy2z 4x2 y2 2xz 2 3yz

A note Method 2 is a way of lsquoworking backwardsrsquo from the way one would multiply

two binomials together

NoticeThe first term of the answer is the product of the (first) variable terms

The last term of the answer is the product of the constant (last) terms

The middle term is the sum of the constant (last) terms

Please keep this in mind as we go forwardhellip

Factoring Polynomials ndash Method 2

How do you factor a trinomial with a leading coefficient of 1

Example Factor x2 ndash 13x + 36

You can use a diamond

The factors are (x ndash 9)(x ndash 4)

Write the middle coefficient here

Write the last term here

ndash13

+36

Now find factors that will multiply to the bottom number and add to the top number

ndash9 ndash4

Example Factor x2 ndash 3x ndash 40 ndash3

ndash40ndash8 +5 The factors are (x ndash 8)(x + 5)

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Factoring Method 1

Factoring polynomials with a Greatest Common Factor (GCF)

IMPORTANT - Always look for a GCF before using any other

factoring method

Steps1) Find the greatest common factor

(GCF)

2) Divide the polynomial by the GCF The quotient (answer from the division) is the other factor

3) Express the polynomial as the product of the quotient and GCF

3cd(2c2 4cd 1)

The answer should look like this

Ex 6c3d 12c2d 2 3cd

Ex 6c3d 12c2d 2 3cd

GCF 3cdSTEP 1

Step2 Divide

(6c3d 12c2d2 3cd) 3cd

2c2 4cd 1

Factor these on your own looking for a GCF

1 6x3 3x2 12x

2 5x2 10x 35

3 16x3y4z 8x2 y2z3 12xy3z 2

3x 2x2 x 4 5 x2 2x 7

4xy2z 4x2 y2 2xz 2 3yz

A note Method 2 is a way of lsquoworking backwardsrsquo from the way one would multiply

two binomials together

NoticeThe first term of the answer is the product of the (first) variable terms

The last term of the answer is the product of the constant (last) terms

The middle term is the sum of the constant (last) terms

Please keep this in mind as we go forwardhellip

Factoring Polynomials ndash Method 2

How do you factor a trinomial with a leading coefficient of 1

Example Factor x2 ndash 13x + 36

You can use a diamond

The factors are (x ndash 9)(x ndash 4)

Write the middle coefficient here

Write the last term here

ndash13

+36

Now find factors that will multiply to the bottom number and add to the top number

ndash9 ndash4

Example Factor x2 ndash 3x ndash 40 ndash3

ndash40ndash8 +5 The factors are (x ndash 8)(x + 5)

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Steps1) Find the greatest common factor

(GCF)

2) Divide the polynomial by the GCF The quotient (answer from the division) is the other factor

3) Express the polynomial as the product of the quotient and GCF

3cd(2c2 4cd 1)

The answer should look like this

Ex 6c3d 12c2d 2 3cd

Ex 6c3d 12c2d 2 3cd

GCF 3cdSTEP 1

Step2 Divide

(6c3d 12c2d2 3cd) 3cd

2c2 4cd 1

Factor these on your own looking for a GCF

1 6x3 3x2 12x

2 5x2 10x 35

3 16x3y4z 8x2 y2z3 12xy3z 2

3x 2x2 x 4 5 x2 2x 7

4xy2z 4x2 y2 2xz 2 3yz

A note Method 2 is a way of lsquoworking backwardsrsquo from the way one would multiply

two binomials together

NoticeThe first term of the answer is the product of the (first) variable terms

The last term of the answer is the product of the constant (last) terms

The middle term is the sum of the constant (last) terms

Please keep this in mind as we go forwardhellip

Factoring Polynomials ndash Method 2

How do you factor a trinomial with a leading coefficient of 1

Example Factor x2 ndash 13x + 36

You can use a diamond

The factors are (x ndash 9)(x ndash 4)

Write the middle coefficient here

Write the last term here

ndash13

+36

Now find factors that will multiply to the bottom number and add to the top number

ndash9 ndash4

Example Factor x2 ndash 3x ndash 40 ndash3

ndash40ndash8 +5 The factors are (x ndash 8)(x + 5)

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

3cd(2c2 4cd 1)

The answer should look like this

Ex 6c3d 12c2d 2 3cd

Ex 6c3d 12c2d 2 3cd

GCF 3cdSTEP 1

Step2 Divide

(6c3d 12c2d2 3cd) 3cd

2c2 4cd 1

Factor these on your own looking for a GCF

1 6x3 3x2 12x

2 5x2 10x 35

3 16x3y4z 8x2 y2z3 12xy3z 2

3x 2x2 x 4 5 x2 2x 7

4xy2z 4x2 y2 2xz 2 3yz

A note Method 2 is a way of lsquoworking backwardsrsquo from the way one would multiply

two binomials together

NoticeThe first term of the answer is the product of the (first) variable terms

The last term of the answer is the product of the constant (last) terms

The middle term is the sum of the constant (last) terms

Please keep this in mind as we go forwardhellip

Factoring Polynomials ndash Method 2

How do you factor a trinomial with a leading coefficient of 1

Example Factor x2 ndash 13x + 36

You can use a diamond

The factors are (x ndash 9)(x ndash 4)

Write the middle coefficient here

Write the last term here

ndash13

+36

Now find factors that will multiply to the bottom number and add to the top number

ndash9 ndash4

Example Factor x2 ndash 3x ndash 40 ndash3

ndash40ndash8 +5 The factors are (x ndash 8)(x + 5)

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Ex 6c3d 12c2d 2 3cd

GCF 3cdSTEP 1

Step2 Divide

(6c3d 12c2d2 3cd) 3cd

2c2 4cd 1

Factor these on your own looking for a GCF

1 6x3 3x2 12x

2 5x2 10x 35

3 16x3y4z 8x2 y2z3 12xy3z 2

3x 2x2 x 4 5 x2 2x 7

4xy2z 4x2 y2 2xz 2 3yz

A note Method 2 is a way of lsquoworking backwardsrsquo from the way one would multiply

two binomials together

NoticeThe first term of the answer is the product of the (first) variable terms

The last term of the answer is the product of the constant (last) terms

The middle term is the sum of the constant (last) terms

Please keep this in mind as we go forwardhellip

Factoring Polynomials ndash Method 2

How do you factor a trinomial with a leading coefficient of 1

Example Factor x2 ndash 13x + 36

You can use a diamond

The factors are (x ndash 9)(x ndash 4)

Write the middle coefficient here

Write the last term here

ndash13

+36

Now find factors that will multiply to the bottom number and add to the top number

ndash9 ndash4

Example Factor x2 ndash 3x ndash 40 ndash3

ndash40ndash8 +5 The factors are (x ndash 8)(x + 5)

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Factor these on your own looking for a GCF

1 6x3 3x2 12x

2 5x2 10x 35

3 16x3y4z 8x2 y2z3 12xy3z 2

3x 2x2 x 4 5 x2 2x 7

4xy2z 4x2 y2 2xz 2 3yz

A note Method 2 is a way of lsquoworking backwardsrsquo from the way one would multiply

two binomials together

NoticeThe first term of the answer is the product of the (first) variable terms

The last term of the answer is the product of the constant (last) terms

The middle term is the sum of the constant (last) terms

Please keep this in mind as we go forwardhellip

Factoring Polynomials ndash Method 2

How do you factor a trinomial with a leading coefficient of 1

Example Factor x2 ndash 13x + 36

You can use a diamond

The factors are (x ndash 9)(x ndash 4)

Write the middle coefficient here

Write the last term here

ndash13

+36

Now find factors that will multiply to the bottom number and add to the top number

ndash9 ndash4

Example Factor x2 ndash 3x ndash 40 ndash3

ndash40ndash8 +5 The factors are (x ndash 8)(x + 5)

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

A note Method 2 is a way of lsquoworking backwardsrsquo from the way one would multiply

two binomials together

NoticeThe first term of the answer is the product of the (first) variable terms

The last term of the answer is the product of the constant (last) terms

The middle term is the sum of the constant (last) terms

Please keep this in mind as we go forwardhellip

Factoring Polynomials ndash Method 2

How do you factor a trinomial with a leading coefficient of 1

Example Factor x2 ndash 13x + 36

You can use a diamond

The factors are (x ndash 9)(x ndash 4)

Write the middle coefficient here

Write the last term here

ndash13

+36

Now find factors that will multiply to the bottom number and add to the top number

ndash9 ndash4

Example Factor x2 ndash 3x ndash 40 ndash3

ndash40ndash8 +5 The factors are (x ndash 8)(x + 5)

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Factoring Polynomials ndash Method 2

How do you factor a trinomial with a leading coefficient of 1

Example Factor x2 ndash 13x + 36

You can use a diamond

The factors are (x ndash 9)(x ndash 4)

Write the middle coefficient here

Write the last term here

ndash13

+36

Now find factors that will multiply to the bottom number and add to the top number

ndash9 ndash4

Example Factor x2 ndash 3x ndash 40 ndash3

ndash40ndash8 +5 The factors are (x ndash 8)(x + 5)

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Try factoring these on your own

1) x2 + 2x ndash 15 =

2) x2 - 10 + 24 =

3) x2 + 6x + 9 =

(x - 3)(x + 5)

(x - 4)(x - 6)

(x + 3)(x + 3) = (x + 3)2

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Some polynomials are already prime there are no factors

Look at x2 + 5x + 1

Are there two numbers that multiply to get 1 and add to get 5

No - There are not two integers that will do this and so this quadratic doesnrsquot factor

Remember this as a possibility when factoring any polynomial

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

How do you factor a trinomial whose leading coefficient is not 1

(3x + 1)(x + 4)

Example Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors

Write factors ofthe first term inthis column

3

1

Write factors ofthe last term inthis column

2

2

Multiple diagonally and addSee if the sum matches themiddle term

2 + 6 ne 13

hellipif not try another combination of factors

3

1

1

4

1 + 12 = 13

( )

( )

These are thecoefficients ofthe factors

Factoring Polynomials ndash Method 3

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Example Factor 6d2 + 33d ndash 63

Remember look for the GCF first

GCF 3

3(2d2 + 11d ndash 21)

Now factor the trinomial (using a T)

3(2d ndash 3)(d + 7)

ndash3 + 14

1

2 ndash3

7

= 11

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Try factoring these on your own

1) 3x2 - 2x ndash 8 =

2) 5x2 ndash 17x + 6 =

3) 4x2 + 10x ndash 6 =

(3x - 4)(x + 2)

(x - 3)(x + 5)

(x - 3)(x + 5)

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes

Closing Notesbull Remember ndash whenever you factor any polynomial

your first step is to see if there is a greatest common factor

bull After completing an initial factorization check to see if any of the factors can be factored further (see example of slide 13)

bull Donrsquot get stuck on a polynomial that cannot be factored ndash there are prime polynomials just as there are prime numbers that cannot be factored Eliminate all possibilities then move on

• Sample Presentation on Factoring Polynomials
• Slide 2
• After this presentation you should be able to
• Factoring a polynomial is the same in concept to factoring a mo
• Why factor polynomials
• Factoring a polynomial means expressing it as a product of othe
• Factoring Method 1
• Steps
• Slide 9
• Slide 10
• Slide 11
• A note Method 2 is a way of lsquoworking backwardsrsquo from the way
• Factoring Polynomials ndash Method 2
• Slide 14
• Some polynomials are already prime there are no factors
• How do you factor a trinomial whose leading coefficient is not
• Example Factor 6d2 + 33d ndash 63
• Slide 18
• Closing Notes