Download 5-3 study guide answers

NAME
5-3
DATE
PERIOD
Study Guide and Intervention
Polynomial Functions
Polynomial Functions
A polynomial of degree n in one variable x is an expression of the form
Polynomial in
One Variable
anx n + an - 1x n - 1 + … + a2 x 2 + a1x + a0 ,
where the coefficients an -1 , an - 2 , an - 3 , …, a0 represent real numbers, an is not zero,
and n represents a nonnegative integer.
The degree of a polynomial in one variable is the greatest exponent of its variable. The
leading coefficient is the coefficient of the term with the highest degree.
Polynomial
Function
A polynomial function of degree n can be described by an equation of the form
P(x ) = anx n + an-1x n - 1 + … + a2x 2 + a1x + a0,
where the coefficients an - 1, an - 2, an - 3, …, a0 represent real numbers, an is not zero,
and n represents a nonnegative integer.
Example 1
What are the degree and leading coefficient of 3x2 - 2x4 - 7 + x3 ?
Rewrite the expression so the powers of x are in decreasing order.
-2x4 + x3 + 3x2 - 7
This is a polynomial in one variable. The degree is 4, and the leading coefficient is -2.
Find f(-5) if f(x) = x3 + 2x2 - 10x + 20.
f(x) = x3 + 2x2 - 10x + 20
f (-5) = (-5)3 + 2(-5)2 - 10(-5) + 20
= -125 + 50 + 50 + 20
= -5
Example 3
g(x) =
g(a - 1) =
=
=
2
Original function
Replace x with -5.
Evaluate.
Lesson X-3
5-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 2
Simplify.
Find g(a2 - 1) if g(x) = x2 + 3x - 4.
x2 + 3x - 4
(a2 - 1)2 + 3(a2 - 1) - 4
a4 - 2a2 + 1 + 3a2 - 3 - 4
a4 + a2 - 6
Original function
Replace x with a2 - 1.
Evaluate.
Simplify.
Exercises
State the degree and leading coefficient of each polynomial in one variable. If it is
not a polynomial in one variable, explain why.
2. 100 - 5x3 + 10 x7 7; 10
3. 4x6 + 6x4 + 8x8 - 10x2 + 20
1. 3x4 + 6x3 - x2 + 12 4; 3
8; 8
4. 4x2 - 3xy + 16y2
not a polynomial in one
variable; contains two variables
5. 8x3 - 9x5 + 4x2 - 36
x2
x6
x3
1
6. −
-−
+−
-−
18
25
1
6; - −
25
5; -9
36
72
Find f(2) and f(-5) for each function.
7. f(x) = x2 - 9
-5; 16
Chapter 5
8. f (x) = 4x3 - 3x2 + 2x - 1
23; -586
17
9. f(x) = 9x3 - 4x2 + 5x + 7
73; -1243
Glencoe Algebra 2
NAME
DATE
5-3
PERIOD
Study Guide and Intervention
(continued)
Polynomial Functions
Graphs of Polynomial Functions
End Behavior of
Polynomial
Functions
If the degree is even and the leading coefficient is positive, then
f(x) → +∞ as x → -∞
f(x) → +∞ as x → +∞
If the degree is even and the leading coefficient is negative, then
f(x) → -∞ as x → -∞
f(x) → -∞ as x → +∞
If the degree is odd and the leading coefficient is positive, then
f(x) → -∞ as x → -∞
f(x) → +∞ as x → +∞
If the degree is odd and the leading coefficient is negative, then
f(x) → +∞ as x → -∞
f(x) → -∞ as x → +∞
The maximum number of zeros of a polynomial function is equal to the degree of the polynomial.
A zero of a function is a point at which the graph intersects the x-axis.
On a graph, count the number of real zeros of the function by counting the number of times the
graph crosses or touches the x-axis.
Real Zeros of a
Polynomial
Function
Example
Determine whether the graph represents an odd-degree polynomial
or an even-degree polynomial. Then state the number of real zeros.
4
As x → -∞, f(x) → -∞ and as x → +∞, f(x) → +∞,
so it is an odd-degree polynomial function.
The graph intersects the x-axis at 1 point,
so the function has 1 real zero.
f (x)
2
-4
-2
4x
2
O
-2
Exercises
For each graph,
a. describe the end behavior,
b. determine whether it represents an odd-degree or an even-degree function, and
c. state the number of real zeroes.
1.
2.
4
3.
f (x)
4
2
-4
-2
O
f(x)
4
2
2
4x
-4
-2
O
f(x)
2
2
4x
-4
-2
O
-2
-2
-2
-4
-4
-4
2
4x
as x → -∞, f(x) → ∞;
as x → -∞, f(x) → -∞;
as x → -∞, f(x) → ∞;
as x → ∞, f(x) → ∞;
as x → ∞, f(x) → -∞;
as x → ∞, f(x) → -∞;
even; 6
even; double zero
odd; 3
Chapter 5
18
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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