(b)
Twice First Last
term term(c)
The first and last terms are perfect squares.
and
Twice the product of the first and last terms of the binomial is
which is notthe middle term of
This trinomial is not a perfect square. In fact, the trinomial cannot be factored even
with the methods of the previous sections. It is a prime polynomial.
(d)
Factor out the common factor, 3z.
is a perfect square trinomial.= 3 z 12 z+ 522 Factor. NOW TRY
= 3 z 312 z 22 + 212 z 2152 + 524 4 z^2 + 20 z+ 25
= 3 z 14 z^2 + 20 z+ 252
12 z^3 + 60 z^2 + 75 z
25 y^2 + 20 y+ 16.
2 # 5 y# 4 = 40 y,
5 y+ 4
25 y^2 = 15 y 22 16 = 42
25 y^2 + 20 y+ 16
9 m^2 - 24 m+ 16 = 13 m 22 + 213 m 21 - 42 + 1 - 422 = 13 m- 422
384 CHAPTER 6 Factoring and Applications
NOW TRY
EXERCISE 5
Factor each trinomial.
(a)
(b)
(c)
(d) 80 x^3 + 120 x^2 + 45 x
9 x^2 + 6 x+ 44 p^2 - 28 p+ 49t^2 - 18 t+ 81NOTE
1. The sign of the second term in the squared binomial is always the same as the sign
of the middle term in the trinomial.
2. The first and last terms of a perfect square trinomial must be positive,because
they are squares. For example, the polynomial cannot be a perfect
square, because the last term is negative.
3. Perfect square trinomials can also be factored by using grouping or the FOIL
method, although using the method of this section is often easier.
x^2 - 2 x- 1
OBJECTIVE 3 Factor a difference of cubes.We can factor a difference of
cubesby using the following pattern.
Factoring a Difference of Cubesx^3 y^3 1 xy 21 x^2 xyy^22
This pattern for factoring a difference of cubes should be memorized.To see that
the pattern is correct, multiply
Multiply vertically.
(Section 5.5)x^3 - y^3 Add.
x^3 +x^2 y+xy^2 x 1 x^2 +xy+y^22
- x^2 y - xy^2 - y^3 - y 1 x^2 +xy+y^22
x -y
x^2 + xy + y^2
1 x-y 21 x^2 + xy+ y^22.
NOW TRY ANSWERS
- (a)
(b)
(c)prime
(d) 5 x 14 x+ 322
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