FACTORINGGCF and Binomials

To factor means to write a number or expression as a product of primes.

In other words, to write a number or expression as things being multiplied together.

The things being multiplied together are called factors.

Here is a simple example of factoring:

The factors are 2, 2 and 3 which are all prime numbers. (They are only divisible by themselves and 1.)

12 2 2 3

Factoring is the opposite of simplifying.

To go from (x+3)(x-6) to 2 6 18x x

2 5 14x x

you simplify.

To go from to(x+7)(x-2)

you factor.

2

(3 2)( 5)

3 13 10

x x

x x

Simplifying Factoring

2 16

( 4)( 4)

x

x x

Simplified. No parentheses and no like terms.

Factored. A product of primes.

2

3 2 2

2 (3 6 )

6 12

x y x y

x y x y

Factored. A product of primes. There are 4 factors.

22, , and (3 6 )x y x y

Simplified. No parentheses and no like terms.

The first thing you always do when factoring is look for a greatest common factor.

GCFGCF: the biggest number or expression that

all the other numbers or expressions can be divided by.

The Process of Factoring

What is the GCF of 27 and 18? The biggest number they are both divisible by is 9

so the GCF is 9.

What is the GCF of 16x2 and 12x? The biggest number that goes into 12 and 16 is 4. The biggest thing x2 and x are divisible by is x. The GCF is 4x.

** If all the terms contain the same variable, the GCF will contain the lowest power of

that variable.

GCF

Factoring out or pulling out the GCF is using the distributive property backwards.

Factoring out the GCF

3 ( 6)x x 23 18x xDistribute 3x

23 18x x 3 ( 6)x x factor out 3x

Factor

Factoring out the GCF

4 3 25 10 25x x x 1. Find the GCF GCF = 5x2

2. Pull out the GCF 5x2(____ - ____ + ____)

3. Divide each term by the GCF to fill in the parentheses.

4 3 2 2 25 10 25 5 ( 2 5)x x x x x x

4 3 22

2 2 2

5 10 252 5

5 5 5

x x xx x

x x x

Distribute to check your answer.

Factor

Factoring out the GCF

2 3 5 2 816 14 4a b a b a b 1. Find the GCF GCF = 2a2b

2. Pull out the GCF 2a2b(____ - ____ + ____)

3. Divide each term by the GCF to fill in the parentheses.

2 3 5 2 8 2 2 3 616 14 4 2a (8 7 2 )a b a b a b b b a b a

2 3 5 2 82 3 6

2 2 2

16 14 48 7 2

2 2 2

a b a b a bb a b a

a b a b a b

Factor

Factoring out the GCF

22 ( 5) 3( 5)x x x 1. Find the GCF GCF = (x + 5)

2. Pull out the GCF (x + 5)(_____ - _____)

3. Divide each term by the GCF to fill in the parentheses.

22 ( 5) 3( 5)

( 5) ( 5)

x x x

x x

22 3x

2 22 ( 5) 3( 5) ( 5)(2 3)x x x x x

Factor

Factoring out the GCF

3 213 10x y1. Find the GCF HMMMM?

These two terms do not have a common factor other than 1!

If an expression can’t be factored it is prime.

You try: Factor

Factoring out the GCF

5 3 47 4m m m 1. Find the GCF 2. Pull out the GCF 3. Divide each term by the GCF

to fill in the parentheses.

5 3 4 3 27 4 ( 7 4 )m m m m m m Better written as-

3 2( 4 7)m m m

You try: Factor

Factoring out the GCF

3 ( 7) 2( 7)x x x 1. Find the GCF 2. Pull out the GCF 3. Divide each term by the GCF

to fill in the parentheses.

3 ( 7) 2( 7) ( 7)(3 2)x x x x x

Factoring Binomials

(2 Terms)

Remember:The first thing you always do when factoring is pull out a GCF if possible.

If there is a GCF and you have factored it out you then look to see if there is any other factoring that can be done.

A sum or difference of cubes involves two terms that are both perfect cubes. The operation between them can be addition or subtraction.

When you have to factor an expression with two terms it could be a difference of squares or a sum or difference of cubes.

A difference of squares involves two terms that are both perfect squares and subtraction.

perfect square – perfect square

Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169…

If a variable has an even exponent then it is a perfect square.

Difference of Squares

2 4 16, , ...x x x

You factor a difference of squares using the following rule:

2 2 ( )( )a b a b a b

Difference of Squares

2a

2b

A difference of squares factors as binomial conjugates. There will be two binomial factors containing the same terms but one will be addition and one will be subtraction.

Factor:

Difference of Squares

24 64x

24x 64

(2 8)(2 8)x x

addition subtraction

Check your answer by FOILing!

Sum of cubes and difference of cubes are both binomials. Both terms are perfect cubes.

perfect cube + perfect cube or

perfect cube – perfect cube

Perfect cubes: 1, 8, 27, 64, 125… or any variable with an exponent divisible by 3.

Sum and Difference of Cubes

A sum or difference of cubes will have two factors. One is a binomial the other is a trinomial.

Sum and Difference of Cubes

Factor a sum or difference of cubes using the following rule:

Sum and Difference of Cubes

a3 b3 a b a2 ab b2 a3 b3 a b a2 ab b2

Notice the terms in the binomial factor are the cubed root of the terms in the original problem and the sign is the same.

You can remember this by remembering CSC (cubed root, same sign, cubed root)

Sum and Difference of Cubes

a3 b3 a b a2 ab b2 a3 b3 a b a2 ab b2

Notice the first and last terms in the trinomial factor are the square of the terms in the binomial factor.

Notice that the middle term in the trinomial factor is the product of the two terms in the binomial factor.

Sum and Difference of Cubes

a3 b3 a b a2 ab b2 a3 b3 a b a2 ab b2

Notice that the first sign in the trinomial factor is the opposite of the sign in the binomial factor.

Notice that the last operation is always addition.

Sum and Difference of Cubes

a3 b3 a b a2 ab b2 a3 b3 a b a2 ab b2

So, you can remember what goes in the binomial factor by remembering CSC

(Cubed root, Same sign, Cubed root)

Sum and Difference of Cubes

a3 b3 a b a2 ab b2 a3 b3 a b a2 ab b2

To remember what goes in the trinomial factor

just remember SOPAS

Product,Opposite sign,Square, SquareAdd,