C H A P T E R 2 Polynomial and Rational Functions

236 Chapter 2 Polynomial and Rational Functions
77. True
79.
x 3 2x 2 11x 12 x 3x 1x 4
The test intervals are , 3, 3, 1, 1, 4, and
4, .
x 2 bx 4 0 80.
To have at least one real solution, b This
occurs when b ≤ 4 or b ≥ 4. This can be written as
, 4 4, .
2 16 ≥ 0.
78. True
The y-values are greater than zero for all values of x.
x 2 bx 4 0
To have at least one real solution,
b 2 414 ≥ 0
b 2 16 ≥ 0.
This inequality is true for all real values of b. Thus, the
interval for b such that the equation has at least one real
solution is , .
81.
To have at least one real solution, b2 3x
4310 ≥ 0.
2 bx 10 0 82. 2x
To have at least one real solution,
2 bx 5 0
b 2 120 ≥ 0
b 120b 120 ≥ 0
Critical numbers: b ±120 ±230
Test intervals:
, 230, 230, 230, 230,
Test: Is b
Solution set: , 230 230,
2 120 ≥ 0?
83. (a) If a > 0 and c ≤ 0, then b can be any real number. If
and then for b to be greater than or
equal to zero, b is restricted to b < 2ac or
b > 2ac.
2 a > 0 c > 0, 4ac
(b) The center of the interval for b in Exercises 79–82 is 0.
85. 4x 2 20x 25 2x 5 2 86.
87.
x 2x 2x 3
b 2 425 ≥ 0
b 2 40 ≥ 0.
This occurs when b ≤ 210 or b ≥ 210. Thus,
the interval for b such that the equation has at least
one real solution is , 210 210, .
84. (a)
(b)
x a, x b
− + +
− − +
+ − +
a
(c) The real zeros of the polynomial
x 3 2 16 x 3 4x 3 4
b
x 7x 1
x2x 3) 4x 3 x2 4x 3 88. 2x 4 54x 2xx3 27
89. Area lengthwidth 90.
2x 1x
2x 2 x
2xx 3x 2 3x 9
Area 1
2baseheight
1
2b3b 2
3
2 b2 b
x