CH. 8 FACTORING
FACTOR (noun) –Any of two or more quantities which form a product when
multiplied together.
12 can be rewritten as 3*4, where 3 and 4 are FACTORS of 12.
FACTOR (verb) - To factor an ex pression is to rewrite it as a product
of 2 or
more quantities. Factoring is sometimes called FACTORIZATION.
12 can be FACTORED into the product 3*4.
Variable Expressions can also be factored.
35x^2 can be factored into 7(5x^2) or 5x(7x) or 5(7x^2), etc…
How about this polynomial expression ?
5x^3 – 35x^2 + 10x
To factor a polynomial, the FIRST STEP is to look for a GREATEST COMMON
FACTOR.
The GCF of a polynomial is the GREATEST number (or variable expression) that
is a factor of every term in the expression . That is, it is the variable
expression
that is the GCF of the coefficients, and the GCF of each of the variables.
How many terms are there?_3__
What are they? _5x^2, -35x^2, 10x__
What is the GCF of these terms?
Let’s take another look at the polynomial: 5x^3 – 35x^2 + 10x
Coefficients:
The Coefficients are 5, -35, and 10. The Coefficient GCF is 5.
Variables:
The only variable is x, the first term has x^3, the 2nd term has x^2, the third
term has x.
The GCF is the greatest x power that can go into ALL of those terms, but
practically this
means it will be the variable term with the smallest power: x
The GCF of 5x^3 – 35x^2 + 10x is 5x
Now to factor the polynomial:
Rewrite the polynomial as a product with 5x as one factor , and the remaining
expression
(after dividing each term by 5x).
Example 2B :
Coefficients :16,8, and -12
GCF : _4__
GCF :__x^2_
GCF :__y_
Put it all together and the GCF of the polynomial is: 4x^2y
NOW YOU TRY:
Factor this:
FACTORING BY GROUPING
Example 3:
y(x + 2) + 3(x + 2)
Remember, a FACTOR is something being multiplied in a product.
Do you a common factor in this expression?
Example 4:
2x(x - 5) + y(5 - x)
At first, it looks like there is no common factor, but notice that
x-5 and 5-x are very similar.
In fact, - (5-x) = -5+x = x-5
So we can rewrite (5-x) has –(x-5)
2x(x - 5) + y(-(x - 5))
= 2x(x - 5) –y(x - 5)
=(x – 5)(2x – y)
Try factoring 3y(5x-2) – 4(2-5x)
Example 5:
If there is not a common factor of ALL the terms, you can
factor by
GROUPING the terms in to groups that DO have a common factor.
Factor out a -2 instead of 2 so that
this group will have a common
factor to the other group (3y – 4).
= (3y - 4)(y^2 – 2)
YOU TRY FACTORING y^5- 5y^3 + 4y^2 - 20
FACTORING POLYNOMIALS OF THE FORM
Example 1:
Factor the polynomial:
STEP 1: Is there a GCF of all three terms? NO
STEP 2: Is this polynomial a trinomial with degree
2? YES
Since this is a trinomial with degree 2, it is possible that this
polynomial can be factored into 2 binomials :
Remember, the FOIL method for
So from this general form, we see that the factors of the
Last Term
(ab), must add up to make the Middle Term’s coefficient.
STEP 3: Try different factor ’s of the Last Term that will add up to
the Middle Term’s coefficient.
Polynomial: x^2 + 18x + 32
Last Term: 32
Middle Term’s Coefficient: 18
FACTORIZATION: (x + 2)(x + 16)
Example 2: Factor x^2 – 6x – 16
STEP 1: Is there a GCF of all three terms? NO
STEP 2: Is it a trinomial with degree 2? YES
FACTORIZATION: (x – 2)(x +8)
UNFACTORABLE TRINOMIALS:
x^2 – 6x – 8
There are no factors of -8 that can add up to -6,
So this is considered a “prime” polynomial and is
“nonfactorable over the integers.”
PROBLEM 3 P. 427
Factor 3a^2b – 18ab – 81b
STEP 1: Is there a GCF of all three terms? YES
GCF is ________
Factor out the GCF:
STEP 2: After factoring out the GCF is one of the
factors a trinomial
with degree 2? YES
STEP 3:
Find factors of the last term of the trinomial that add up to the middle
term’s coefficient and factor into two binomials.
STEP 4: Don’t forget STEP 1’S GCF in you final
factorization!
EXAMPLE 4:
Factor x^2 + 9xy + 20y^2
STEP 1: Is there a GCF of all terms? NO
STEP 2: Is this a trinomial with degree 2? YES
STEP 3: Find factors of the last term of the
trinomial that add up
to the middle term’s coefficient and factor into two binomials.
In this case, make sure the factors of the the last term are “like
terms” that can be combined. (eg. 20 and 1y^2 are not like terms
but are factors of 20y^2)
Factors of Last
Term, 20y^2 |
Sum of Those
Factors |
| 20y, 1y |
20y+1y =21y |
| 2y, 10y |
2y + 10y = 12y |
| 4y, 5y |
4y + 5y = 9y |
FACTORIZATION: (x + 4y)(x + 5y)
