8-1 and 8-2 factoring using the distributive property algebra 1 glencoe mcgraw-hilllinda stamper

8-1 and 8-2 Factoring Using the Distributive

Property

Algebra 1 Glencoe McGraw-Hill Linda Stamper

You have used the distributive property to determine a product – for example:

2xx5

You can also use the distributive property to take the product and return it to factored form – for example: x10x5 2

Today you will use the distributive property to factor out constants and or variables that are common terms of a polynomial.A polynomial is prime if it cannot be factored using integer coefficients. To factor a polynomial completely, write it as the product of a monomial and prime factors.

x10x5 2

2xx5

Find the greatest monomial factor. Then factor it out of the expression. x8x2 2 Write as prime factors. xx2 Circle common primesFind the GMF (multiply the common primes).Use the distributive property to factor out the GMF

Check – Multiply the factors together using the distributive property.

x8x24xx2 2

x222

x2x 4

Whatever is NOT circled

goes in parenthese

s.

Find the greatest monomial factor. Then factor it out of the expression. x8x2 2 The problem.Think of the GCF.Use the distributive property to factor out the GCF

x2 x 4

2x2x2

2x8x

You are using division when you factor the GCF out of the

expression!

Find the greatest monomial factor. Then factor it out of the expression. ab9a3 3 Write as prime factors. aaa3 Circle common primesFind the GMF (multiply the common primes).Use the distributive property to factor out the GMF


ab9a3b3aa3 32

ba33

a3

2a 3b

Whatever is NOT circled

goes in parenthese

s.

Find the greatest monomial factor. Then factor it out of the expression. ba9a3 3 The problem.Think of the GMF.Use the distributive property to factor out the GMF

a3 2a 3b

3aa3 3

3a9ab

Find the greatest monomial factor. Then factor it out of the expression.

Write as prime factors. m55 Circle common primes

n37

prime

n21m25


23 x211x33


Example 1Example 2

x3x6 2

Example 1 Find the greatest monomial factor. Then factor it out of the expression.

23 x211x33 xxx113


232 x121x3311x3x11

xx1111

211x3x 11 ( )

23 x21133x 211x x3 11

23

11x33x

22

11x121x

Example 2 Find the greatest monomial factor. Then factor it out of the expression.

x3x6 2 xx32


x3x61x2x3 2

x3

3x2x 1 ( )

3x6x2

3x x2 1

3x6x2

3x3x

Example 3

23 21x14x


Example 4Example 5Example 6

30x25x5x 23

24x20x4x 23

2n4n2n 23

32x7x2

6x5x5x 2

6x5x4x 2

1n2n2n 2

Using the distributive property to factor polynomials having four or more terms is called factoring by grouping because pairs of terms are grouped together and factored. The distributive property is then applied a second time to factor a common binomial factor. 6a3b8ab4 Group terms with common factors.Factor the GMF from each group.Factor the common binomial factor.

Check – Multiply the factors together using FOIL.

6a3b8ab4

b4 2a 3 2a

2a

b4 3

The problem.

2a

Sometimes you can group terms in more than one way when factoring a polynomial. Here is an alternate way to group the previous problem.

6a3b8ab4 Group terms with common factors.Factor the GMF from each group.Factor the common binomial factor.

Notice that this result is as the previous grouping.

6b8a3ab4

a 3b4 2 2b4

2b4

The problem.

Factor the polynomial.

20x8x15x6 2


Example 7

Example 8 5xy5xy

7y2x7xy2 Example 9

Example 10 21y3xy5x35

5xx

4x3 4x3


20x8x15x6 2


20x8x15x6 2

5x2 5x2

5x2

Example 7

Example 8

4x35x2

Undo doubl

e sign!

5

5xy5xy

5y5xxy

1y 1y

1y

7y2

7y2


7y2x7xy2


7x7y2xy2

1x 1x

1x

Example 9

Example 10 21y3xy5x35

21y3xy5xy35

x5 y7 3 7y

x5 1 7y 3 7y

7y37yx5

3x57y

8-A3 Page 423 # 19–27, and Page 429 # 9–20.

8-1 and 8-2 factoring using the distributive property algebra 1 glencoe mcgraw-hilllinda stamper

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