College Algebra 9th txtbk

26 CHAPTER R BASIC ALGEBRAIC OPERATIONS
A nonprime polynomial is said to be factored completely relative to a given set of
numbers if it is written as a product of prime polynomials relative to that set of numbers.
In Examples 5 and 6 we review some of the standard factoring techniques for polynomials
with integer coefficients.
EXAMPLE 5 Factoring Out Common Factors
Factor out, relative to the integers, all factors common to all terms:
(A) 2x 3 y 8x 2 y 2 6xy 3 (B) 2x(3x 2) 7(3x 2)
SOLUTIONS (A) 2x 3 y 8x 2 y 2 6xy 3 (2xy)x 2 (2xy)4xy (2xy)3y 2 Factor out 2xy.
2xy(x 2 4xy 3y 2 )
(B) Factor out 3x 2.
2x(3x 2) 7(3x 2) 2x(3x 2) 7(3x 2)
(2x 7)(3x 2)
MATCHED PROBLEM 5
Factor out, relative to the integers, all factors common to all terms:
(A)
3x 3 y 6x 2 y 2 3xy 3
(B)
3y(2y 5) 2(2y 5)
The polynomials in Example 6 can be factored by first grouping terms to find a common
factor.
EXAMPLE 6 Factoring by Grouping
Factor completely, relative to the integers, by grouping:
(A) 3x 2 6x 4x 8
(C) 3ac bd 3ad bc
(B) wy wz 2xy 2xz
SOLUTIONS
(A) 3x 2 6x 4x 8
(B)
(C)
(3x 2 6x) (4x 8)
3x(x 2) 4(x 2)
(3x 4)(x 2)
wy wz 2xy 2xz
(wy wz) (2xy 2xz)
w(y z) 2x(y z)
(w 2x)(y z)
Group the first two and last two terms.
Remove common factors from each group.
Factor out the common factor (x 2).
Group the first two and last two terms—be careful of signs.
Remove common factors from each group.
Factor out the common factor ( y z).
3ac bd 3ad bc
In parts (A) and (B) the polynomials are arranged in such a way that grouping the
first two terms and the last two terms leads to common factors. In this problem neither
the first two terms nor the last two terms have a common factor. Sometimes
rearranging terms will lead to a factoring by grouping. In this case, we interchange