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Factoring

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Factoring

Polynomials

Factoring a polynomial means expressing it as a product of other polynomials.

Factoring Method #1

Factoring polynomials with a common monomial factor (using GCF).

**Always look for a GCF before using any other factoring method.

Steps:

1. Find the greatest common factor (GCF).

2. Divide the polynomial by the GCF. The quotient is the other factor.

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

Step 1:

Step 2: Divide by GCF

The answer should look like this:

Factor these on your own looking for a GCF.

Factoring Method #2

Factoring polynomials that are a difference of squares.

A “Difference of Squares” is a binomial (*2 terms only*) and it factors like this:

To factor, express each term as a square of a monomial then apply the rule...

Here is another example:

Try these on your own:

Sum and Difference of Cubes:

Apply the rule for sum of cubes:

Write each monomial as a cube and apply either of the rules.

Rewrite as cubes

Apply the rule for difference of cubes:

Rewrite as cubes

Factoring a trinomial in the form:

where a = 1

Factoring Method #3

Factoring a trinomial:

1. Write two sets of parenthesis, (x )(x). These will be the factors of the trinomial.

2. Find the factors of the c term that add to

the b term. For instance, let

c = d·eand d+e = b

then the factors are

(x+d)(x+e ).

.

Next

x

x

-2

-4

1 + 8 = 9

2 + 4 = 6

-1 - 8 = -9

-2 - 4 = -6

Factors of +8: 1 & 8

2 & 4

-1 & -8

-2 & -4

Factors of +8 that add to -6

F

O

I

L

Check your answer by using FOIL

Lets do another example:

Don’t Forget Method #1.

Always check for GCF before you do anything else.

Find a GCF

Factor trinomial

When a>1, let’s do something different!

Step 1:

Multiply a · c

= - 30

Step 2: Find the factors of a·c (-30) that add to the b term

Factors of 6 · (-5) :1, -30 1+-30 = -29

-1, 30 -1+30 = 29

2, -15 2+-15 =-13

-2, 15 -2+15 =13

3, -10 3+ -10 =-7

-3, 10 -3+ 10 =7

5, -6 5+ -6 = -2

-5, 6 -5+6 =1

Step 2: Find the factors of a·c that add to the b term

Let a·c = d and d = e·f

then e+f = b

d = -30

e = -2

f = 15

-2, 15 -2+15 =13

Step 3: Rewrite the expression separating

the b term using the factors e and f

-2x+15x -5

13x

Step 4: Group the first

two and last two terms.

-2x + 15x -5

Step 4: Group the first

Two and last two terms.

Step 5: Factor GCF

from each group

Check!!!! If you cannot

find two common factors,

Then this method does

not work.

Step 6: Factor out GCF

- 2x + 15x - 5

3x - 1)+ 5(3x - 1)

Common factors

3x - 1)(2x + 5)

I am not a

fan of guess

and check!

Try:

F

O

I

L

Step 3: Place the factors inside the parenthesis until O + I = bx.

This doesn’t work!!

O + I = 30 x - x = 29x

F

O

I

L

Switch the order of the second terms

and try again.

This doesn’t work!!

O + I = -6x + 5x = -x

F

O

I

L

O+I = 15x - 2x =13x

IT WORKS!!

Try another combination:

Switch to 3x and 2x

Factoring Technique #3

continued

Factoring a perfect squaretrinomial in the form:

Perfect Square Trinomials can be factored just like other trinomials (guess and check), but if you recognize the perfect squares pattern, follow the formula!

b

a

Does the middle term fit the pattern, 2ab?

Yes, the factors are (a + b)2 :

b

a

Does the middle term fit the pattern, 2ab?

Yes, the factors are (a - b)2 :

Factoring Technique #4

Factoring By Grouping

for polynomials

with 4 or more terms

1. Group the first set of terms and last set of terms with parentheses.

2. Factor out the GCF from each group so that both sets of parentheses contain the same factors.

3. Factor out the GCF again (the GCF is the factor from step 2).

Factoring By Grouping

Example 1:

Step 1: Group

Step 2: Factor out GCF from each group

Step 3: Factor out GCF again

Example 2:

Try these on your own:

Answers: