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
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: