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

258 Chapter 2 Polynomial and Rational Functions
151. The maximum (or minimum) value of a quadratic
function is located at its graph’s vertex. To find the
vertex, either write the equation in standard form or
use the formula
1.
fx ax 3 bx 2 cx d
x k) ax3 bx2 ax
cx d
2 ak bx ak2 bk c
ax 3 akx 2
ak bx 2 cx
ak bx 2 ak 2 bkx
ak 2 bk cx d
ak 2 bk cx ak 3 bk 2 ck
ak 3 bk 2 ck d
Thus, and
Since the remainder r ak f k r.
3 bk2 f k ak ck d,
3 bk2 f x ax
ck d.
3 bx2 cx d x kax2 ak bx ak2 bx c ak3 bk2 ck d
2. (a)
b b
, f
2a 2a .
If the leading coefficient is positive, the vertex is a
minimum. If the leading coefficient is negative, the
vertex is a maximum.
Problem Solving for Chapter 2
(b)
(c)
y
y 3 y 2
1 2
2 12
3 36
4 80
5 150
6 252
7 392
8 576
9 810
10 1100
x
2 3
x
2 2
36 ⇒ x
1
8
3 ⇒ x 6
2 x3 1
8 2x2 1
8 288
x3 2x2 288; a 1, b 2 ⇒ a2
x
1
b3 8
3 x2 252 ⇒ x 6
152. Answers will vary. Sample answer:
Polynomials of degree n > 0
with real coefficients can
be written as the product of linear and quadratic factors
with real coefficients, where the quadratic factors have
no real zeros.
153. An asymptote of a graph is a line to which the graph becomes arbitrarily close as x increases or decreases without bound.
(d)
(e)
(f)
(g)
Setting the factors equal to zero and solving for the
variable can find the zeros of a polynomial function.
To solve an equation is to find all the values of the
variable for which the equation is true.
3x3 x2 90; a 3, b 1 ⇒ a2
b3 9
93x 3 9x 2 990
3x 3 3x 2 810 ⇒ 3x 9 ⇒ x 3
2x3 5x2 2500; a 2, b 5 ⇒ a2 4
b3
125
4
125 2x3 4
125 5x2 4
125 2500
2x
5 3
2x
5 2
80 ⇒ 2x
4 ⇒ x 10
5
7x3 6x2 1728; a 7, b 6 ⇒ a2 49
b3 216
49
216 7x3 49
216 6x2 49
216 1728
7x
6 3
7x
6 2
392 ⇒ 7x
7 ⇒ x 6
6
10x3 3x2 297; a 10, b 3 ⇒ a2 100
b3 27
100
27 10x3 100
27 3x2 100
27 297
10x
3 3
10x
3 2
1100 ⇒ 10x
10 ⇒ x 3
3