factoring

B. deTreville HSHS

FACTORING

B. deTreville HSHS

To check your answer to a factoring problem you simplify it by multiplying out the factors.

The expression can be factored as You can check that this is correct by foiling.

The expression can be factored as You can check this by distributing.

You need to recognize when an expression is in factored form and when it is in simplified or polynomial form. Factored form means there are things being multiplied together. Simplified or polynomial form means there are no parentheses and all like terms are combined.

Simplified/Polynomial Factored Factored Simplified/Polynomial

Furthermore, you need to recognize when an expression is not completely factored. If an expression is not completely factored that means there is more factoring that can be done.

2 2 15x x ( 5)( 3)x x

2

2

( 5)( 3)

3 5 15

2 15

x x

x x x

x x

2 4 4 ( 2)( 2)x x x x 22 (5 9) 2 18xy x x y y

Factoring means to write an expression as a product of primes.

2 3 225 15x y xy 25 (5 3)xy xy

AFTER YOU FACTOR L K AT THE FACTORS TO SEE IF THEY CAN STILL BE FACTORED!!!

Whenever you do factoring problems you should ask a series of questions.

Use the following slides to walk you through the factoring process. Use these slides for each problem until you can work through them on your own.

Question 1: Is there a GCF?

The first thing you must do when factoring any expression is pull out the GCF if there is one.

GCF

Don’t forget…… if you pull

out a GCF it must be part

of the final answer. It IS one

of the factors in the answer.

Question 2:

How many terms does the expression have?

2

3

4

Are both terms perfect squares with subtraction between them? Yes

Are both terms perfect squares with addition between them? Yes

Are both terms perfect cubes with subtraction between them? Yes

Are both terms perfect cubes with addition between them? Yes

Start New Problem

DIFFERENCE OF SQUARES

The difference of squares is easy to factor. The factors are as follows:

Ex:

( first term last term)( first term last term)

2 2 factors as ( )( )a b a b a b

Start New Problem

Sum of Squares

A sum of squares cannot be factored. The binomial is prime.

Start New Problem

Difference of CubesRemember: CSC SOPAS

The difference of cubes factors as:

3 31st Term 2nd Term (Square + Product + Square) C S C S O P A S

u a u q p r d q

b m b u p o d u

e e e a o d a

R S R r s u r

o i o e i c e

o g o t t

t n t e

S

i

g

n

Difference of Cubes

3 3 2 2( )( )a b a b a ab b cube root

cube root

square

square

product

Difference of Cubes

3 3 2 2( )( )a b a b a ab b

Same sign Opposite sign

Addition

Start New Problem

Sum of CubesRemember: CSC SOPAS

The sum of cubes factors as:

3 31st Term 2nd Term (Square - Product + Square) C S C S O P A S

u a u q p r d q

b m b u p o d u

e e e a o d a

R S R r s u r

o i o e i c e

o g o t t

t n t e

S

i

g

n

Sum of Cubes

3 3 2 2( )( )a b a b a ab b cube root

cube root

square

square

product

Sum of Cubes

3 3 2 2( )( )a b a b a ab b

Same sign Opposite sign

Addition

Start New Problem

Trinomials

Is the leading coefficient 1? Yes No

TrinomialsIf the leading coefficient

is 1 you can factor the trinomial quick and easy.

2ax bx c factors as ( ___ )( ___ )x x

sum = bproduct = c

Start New Problem

TrinomialsIf the trinomial has a

leading coefficient other than 1 you will use the multiply divide method.

23 16 12x x

Start New Problem

2 16 36x x ( 2)( 18)x x

2 18( )( )

3 3x x

(3 2)( 6)x x

1. Multiply a and c to make a trinomial with a leading coefficient of 1.

2. Factor the new trinomial using the quick and easy method.

3. Divide each constant in both factors by a.

4. Reduce any fractions and make any denominator a coefficient on the variable.

Factor by GroupingGroup the first two and

last two terms using parentheses.

Pull the GCF out of each group.

Pull out the new GCF.

Ex.

( ) ( )ax bx ay by ax ax ay by

( ) ( )x a b y a b ( )( )x y a b Start New Problem

L K

Are you sure you are finished? Can any of the factors still be factored?

If so, factor them. If not then you are ready to start a new problem.

Start New Problem