CHAPTER 8: FACTORING

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CHAPTER 8 FACTORING FACTOR (noun) Any of two
or more quantities which form a product when
multiplied together. 12 can be rewritten as 34,
where 3 and 4 are FACTORS of 12. FACTOR (verb) -
To factor an expression is to rewrite it as a
product of 2 or more quantities. Factoring is
sometimes called FACTORIZATION. 12 can be
FACTORED into the product 34. Variable
Expressions can also be factored. 35x2 can be
factored into 7(5x2) or 5x(7x) or 5(7x2),
etc How about this polynomial expression? 5x3
35x2 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?___ What are
they? _______________ What is the GCF of these
terms? ______ Lets take another look at the
polynomial 5x3 35x2 10x Coefficients The
Coefficients are 5, -35, and 10. The
Coefficient GCF is 5. Variables The only
variable is x, the first term has x3, the 2nd
term has x2, 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 5x3 35x2 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).
This polynomial cannot be factored any more. It
is prime. Well see why later.
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___
___
__
Put it all together and the GCF of the polynomial
is ____ NOW YOU TRY Factor this
6x4y2 9x3y2 12x2y4
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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? y(x 2) 3(x
2) 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) -5x 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)
COMMON FACTOR
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Example 5 3y3 4y2 6y 8 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. (3y3 4y2) ( 6y
8) y2(3y - 4) -2(3y 4) (3y - 4)(y2
2) YOU TRY FACTORING y5- 5y3 4y2 - 20
Factor out a -2 instead of 2 so that this group
will have a common factor to the other group (3y
4).
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FACTORING POLYNOMIALS OF THE FORM x2 bx c
Example 1 Factor the polynomial x2 18x
32 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 ( x a )( x b
) Remember, the FOIL method for (x a)(x b)
x2 ax bx ab x2 (a b)x ab So
from this general form, we see that the factors
of the Last Term (ab), must add up to make the
Middle Terms coefficient. STEP 3 Try different
factors of the Last Term that will add up to the
Middle Terms coefficient. Polynomial x2
18x 32 Last Term __ Middle Terms
Coefficient __
Factors of Last Term Sum of Those Factors
1, 32 132 33
4,8 48 12
2,16 216 18
FACTORIZATION (x 2)(x 16)
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Example 2 Factor x2 6x 16 STEP 1 Is there
a GCF of all three terms? __ STEP 2 Is it a
trinomial with degree 2? __ Last Term __Middle
Terms Coefficient __
Factors of Last Term Sum of Those Factors
1, -16 1 -16 -15
-1, 16 -1 16 15
2, -8 2 -8 -6
-2, 8 -2 8 6
4, -4 4 -4 0
-4,4 -4 4 0
FACTORIZATION (x 2)(x 8)
UNFACTORABLE TRINOMIALS x2 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.
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PROBLEM 3 P. 427 Factor 3a2b 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? ___ STEP
3 Find factors of the last term of the trinomial
that add up to the middle terms coefficient and
factor into two binomials. STEP 4 Dont
forget STEP 1S GCF in you final
factorization!
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EXAMPLE 4 Factor x2 9xy 20y2 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 terms 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 1y2
are not like terms but are factors of
20y2) FACTORIZATION (x 4y)(x 5y)
Factors of Last Term, 20y2 Sum of Those Factors
20y, 1y 20y1y 21y
2y, 10y 2y 10y 12y
4y, 5y 4y 5y 9y