Section 8–6
◆
219
The Perfect Square Trinomial
29. An object is thrown into the air with an initial velocity of 23 m/s. To find the
approximate time it takes for the object to reach a height of 12 m, we must solve the
quadratic equation
Width
h
5t 2 23t 12 0
Factor the left side of this equation.
30. To find the depth of cut h needed to produce a flat of a certain width on a 1-cm-radius
bar (Fig. 8–6), we must solve the equation
4h2 8h 3 0
1 cm
Factor the left side of this equation.
FIGURE 8–6
8–6
The Perfect Square Trinomial
The Square of a Binomial
In Chapter 2, we saw that the expression obtained when a binomial is squared is called a perfect
square trinomial.
◆◆◆
Example 36: Square the binomial (2x 3).
Solution:
(2x 3)2 4x2 6x 6x 9
4x2 12x 9
◆◆◆
Note that in the perfect square trinomial obtained in Example 36, the first and last terms are
the squares of the first and last terms of the binomial.
square
(2x + 3)2
square
4x2 12x 9
The middle term is twice the product of the terms of the binomial.
(2x 3)2
product
6x
twice the product
4x2 12x 9
Also, the constant term is always positive. In general, the following equations apply:
Perfect
Square
Trinomials
(a b)2 a2 2ab b2
47
(a b)2 a2 2ab b2
48
Factoring a Perfect Square Trinomial
We can factor a perfect square trinomial in the same way we factored the general quadratic
trinomial in Sec. 8–5. However, the work is faster if we recognize that a trinomial is a perfect
square. If it is, its factors will be the square of a binomial. The terms of that binomial are the
square roots of the first and last terms of the trinomial. The sign in the binomial will be the same
as the sign of the middle term of the trinomial.
Any quadratic trinomial can be
manipulated into the form of a
perfect square trinomial by a
procedure called completing the
square. We will use that method
in Sec. 14–2 to derive the
quadratic formula.
220
Chapter 8
◆◆◆
◆
Factors and Factoring
Example 37: Factor a2 4a 4.
Solution: The first and last terms are both perfect squares, and the middle term is twice the
product of the square roots of the first and last terms. Thus the trinomial is a perfect square.
Factoring, we obtain
a2 4a 4
square
root
square
root
same
(a 2)2
◆◆◆
9x2y2 6xy 1 (3xy 1)2
Common
Error
Exercise 6
◆
(a b)2 a2 b2
The Perfect Square Trinomial
Factor completely.
1.
3.
5.
7.
9.
11.
13.
15.
x2 4x 4
y2 2y 1
2y2 12y 18
9 6x x2
9x2 6x 1
9y2 18y 9
16 16a 4a2
49a2 28a 4
2.
4.
6.
8.
10.
12.
14.
x2 30x 225
x2 2x 1
9 12a 4a2
4y2 4y 1
16x2 16x 4
16n2 8n 1
1 20a 100a2
Perfect Square Trinomials with Two Variables
Factor completely.
16.
18.
20.
22.
a2 2ab b2
a2 14ab 49b2
a2 10ab 25b2
c2 6cd 9d 2
17.
19.
21.
23.
x2 2xy y2
a2w2 2abw b2
x2 10ax 25a2
x2 8xy 16y2
Expressions That Can Be Reduced to Perfect Square Trinomials
Factor completely.
24.
25.
26.
27.
28.
29.
30.
31.
◆◆◆
Example 38:
1 2x2 x4
z6 16z3 64
36 12a2 a4
49 14x3 x6
a2b2 8ab3 16b4
a4 2a2b2 b4
16a2b2 8ab2c2 b2c4
4a2n 12anbn 9b2n
◆◆◆
Section 8–7
8–7
◆
221
Sum or Difference of Two Cubes
Sum or Difference of Two Cubes
Definition
An expression such as
x3 27
is called the sum of two cubes (x3 and 33). In general, when we multiply the binomial (a b)
and the trinomial (a2 ab b2), we obtain
(a b)(a2 ab b2) a3 a2b ab2 a2b ab2 b3
a3 b3
All but the cubed terms drop out, leaving the sum of two cubes.
Sum of Two
Cubes
a3 b3 (a b)(a2 ab b2)
42
Difference of
Two Cubes
a3 b3 (a b)(a2 ab b2)
43
When we recognize that an expression is the sum (or difference) of two cubes, we can write
the factors immediately.
◆◆◆
Example 39: Factor x3 27.
Solution: This expression is the sum of two cubes, x3 33. Substituting into Eq. 42, with a x
and b 3, yields
x3 27 x3 33 (x 3)(x2 3x 9)
same
sign
opposite
sign
◆◆◆
always ◆◆◆
Example 40: Factor 27x3 8y3.
Solution: This expression is the difference of two cubes, (3x)3 (2y)3. Factoring gives us
27x3 8y3 (3x)3 (2y)3 (3x 2y)(9x2 6xy 4y2)
same
sign
always opposite
sign
Common
Error
◆◆◆
The middle term of the trinomials in Eqs. 42 and 43 is often
mistaken as 2ab.
a3 b3 (a b)(a2 2ab b2)
no!