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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:.

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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




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:



Here is another example: apply the rule...


Try these on your own: apply the rule...


End of day 1
End of Day 1 apply the rule...


Sum and Difference of Cubes: apply the rule...


Apply the rule for sum of cubes: apply the rule...

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

Rewrite as cubes


Apply the rule for difference of cubes: apply the rule...

Rewrite as cubes


Factoring a apply the rule... trinomial in the form:

where a = 1

Factoring Method #3


Factoring a trinomial apply the rule... :

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 apply the rule...

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 apply the rule...

O

I

L

Check your answer by using FOIL


Lets do another example: apply the rule...

Don’t Forget Method #1.

Always check for GCF before you do anything else.

Find a GCF

Factor trinomial


When apply the rule... 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 apply the rule... 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 apply the rule... , 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 apply the rule...

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 apply the rule...

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 apply the rule...

O

I

L

Switch the order of the second terms

and try again.

This doesn’t work!!

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


F apply the rule...

O

I

L

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

IT WORKS!!

Try another combination:

Switch to 3x and 2x


Factoring Technique #3 apply the rule...

continued

Factoring a perfect squaretrinomial in the form:


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


b apply the rule...

a

Does the middle term fit the pattern, 2ab?

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


b apply the rule...

a

Does the middle term fit the pattern, 2ab?

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


Factoring Technique #4 apply the rule...

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: with parentheses.

Step 1: Group

Step 2: Factor out GCF from each group

Step 3: Factor out GCF again


Example 2: with parentheses.


Try these on your own: with parentheses.


Answers: with parentheses.


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