For example,
The following conditions must be true for a binomial to be a difference of squares.
1.Both terms of the binomial must be squares, such as
,,
2.The terms of the binomial must have different signs (one positive and one
negative).
x^2 , 9 y^2 = 13 y 22 , 25 = 52 1 = 12 m^4 = 1 m^222.
= 1 m+ 421 m- 42.
=m^2 - 42
m^2 - 16
SECTION 6.4 Special Factoring Techniques 381
OBJECTIVES
Special Factoring Techniques
6.4
1 Factor a difference
of squares.
2 Factor a perfect
square trinomial.
3 Factor a difference
of cubes.
4 Factor a sum of
cubes.By reversing the rules for multiplication of binomials from Section 5.6,we get rules
for factoring polynomials in certain forms.
Factoring a Difference of Squaresx^2 y^2 1 xy 21 xy 2
NOW TRY
EXERCISE 1
Factor each binomial if
possible.
(a)x^2 - 100 (b)x^2 + 49
Factoring Differences of SquaresFactor each binomial if possible.
(a) (b)
(c)
Because 8 is not the square of an integer, this binomial does not satisfy the con-
ditions above. It is a prime polynomial.
(d)
Since is a sumof squares, it is not equal to Also, we
use FOIL and try the following.
Thus, p^2 + 16 is a prime polynomial. NOW TRY
= p^2 + 8 p+ 16, not p^2 + 16.
1 p+ 421 p+ 42
= p^2 - 8 p+ 16, not p^2 + 16.
1 p- 421 p- 42
p^2 + 16 1 p+ 421 p- 42.
p^2 + 16
x^2 - 8
a^2 - 49 = a^2 - 72 = 1 a+ 721 a- 72 y^2 - m^2 = 1 y+m 21 y- m 2
x^2 - y^2 = 1 x+ y 21 x- y 2EXAMPLE 1
OBJECTIVE 1 Factor a difference of squares.The formula for the product of
the sum and difference of the same two terms is
Reversing this rule leads to the following special factoring rule.
1 x+y 21 x-y 2 =x^2 - y^2.
CAUTION AsExample 1(d)suggests, after any common factor is removed,
a sum of squares cannot be factored.
NOW TRY ANSWERS
- (a)
(b)prime
1 x+ 1021 x- 102