2. Exponents, Polynomials, and Polynomial Functions
Chapter 6
Exponents, Polynomials, and
Polynomial Functions

3. 6.3
Polynomial Functions

4. 6.3 Polynomial Functions Objectives
Recognize and evaluate polynomial functions.
Use a polynomial function to model data.
Add and subtract polynomial functions.
Graph basic polynomial functions.
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5. Definition of a Polynomial Function
6.3 Polynomial Functions
Definition of a Polynomial Function
Polynomial Function
A polynomial function of degree n is defined by
f (x) = an xn an – 1 xn – · · · a1 x a0 ,
for real numbers an,an – 1, , a1, and a0 , where an ≠ 0 and n is a whole
number.
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6. Evaluating Polynomial Functions
EXAMPLE 1
Evaluating Polynomial Functions
Let f(x) = 4x3 – 5x Find each value.
(a) f(2)
f(x) = 4x – 5x
f(2) = 4 • 23 – 5 •
= 4 • 8 – 5 •
= – =
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7. Evaluating Polynomial Functions
EXAMPLE 1
Evaluating Polynomial Functions
Let f(x) = 4x3 – 5x Find each value.
(b) f(–3)
f(x) = 4x – x
f(–3) = 4 • (–3)3 – 5 • (–3)
= 4 • (–27) – •
= – –
= –146
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8. 6.3 Polynomial Functions Functions
While f is the most common letter used to represent functions, recall that
other letters such as g and h are also used. The capital letter P is often used
for polynomial functions.
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9. Using a Polynomial Model to Approximate
6.3 Polynomial Functions
EXAMPLE 2
Using a Polynomial Model to Approximate
Data
The number of U.S. households estimated to see and pay at least one bill
on-line each month during the years 2000 through 2006 can be modeled by
the polynomial function defined by
P(x) = x x ,
where x = 0 corresponds to the year 2000, x = 1 corresponds to 2001, and
so on, and P(x) is in millions. Use this function to approximate the number
of households expected to pay at least one bill on-line each month in 2006.
Since x = 6 corresponds to 2006, we must find P(6).
P(x) = x x
P(6) = (6) (6)
Let x = 6. =
Evaluate.
Thus, in 2006 about million households are expected to pay at least
one bill on-line each month.
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10. Adding and Subtracting Functions
6.3 Polynomial Functions
Adding and Subtracting Functions
Adding and Subtracting Functions
If f(x) and g(x) define functions, then
(f + g) (x) = f (x) g(x) Sum function
and (f – g) (x) = f (x) – g(x). Difference function
In each case, the domain of the new function is the intersection of the
domains of f(x) and g(x).
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11. Adding and Subtracting Functions
6.3 Polynomial Functions
EXAMPLE 3
Adding and Subtracting Functions
For the polynomial functions defined by
f(x) = 2x2 – 3x and g(x) = x x – 5,
find (a) the sum and (b) the difference.
(a) (f + g) (x) = f (x) g(x) Use the definition.
= (2x2 – 3x + 4) + (x x – 5) Substitute.
= 3x x – 1 Add the polynomials.
(b) (f – g) (x) = f (x) – g(x) Use the definition.
= (2x2 – 3x + 4) – (x x – 5) Substitute.
= (2x2 – 3x + 4) + (–x2 – 9x + 5) Change subtraction
to addition.
= x2 – 12x Add.
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12. Adding and Subtracting Functions
6.3 Polynomial Functions
EXAMPLE 4
Adding and Subtracting Functions
For the polynomial functions defined by
f(x) = 4x2 – x and g(x) = 3x,
find each of the following.
(a) (f + g) (5)
(f + g) (5) = f (5) g(5) Use the definition.
= [4(5)2 – 5] (5) Substitute.
= 110
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13. Adding and Subtracting Functions
6.3 Polynomial Functions
EXAMPLE 4
Adding and Subtracting Functions
For the polynomial functions defined by
f(x) = 4x2 – x and g(x) = 3x,
find each of the following.
(a) (f + g) (5)
Alternatively, we could first find (f + g) (x).
(f + g) (x) = f (x) g(x) Use the definition.
= (4x2 – x) x Substitute.
= 4x x
Then,
(f + g) (5) = 4(5) (5) = The result is the same.
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14. Adding and Subtracting Functions
6.3 Polynomial Functions
EXAMPLE 4
Adding and Subtracting Functions
For the polynomial functions defined by
f(x) = 4x2 – x and g(x) = 3x,
find each of the following.
(b) (f – g) (x) and (f – g) (3)
(f – g) (x) = f (x) – g(x) Use the definition.
= (4x2 – x) – 3x Substitute.
= 4x2 – 4x Combine like terms.
Then,
(f – g) (3) = 4(3)2 – 4(3) = Substitute.
Confirm that f (3) – g(3) gives the same result.
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15. Basic Polynomial Functions
The simplest polynomial function is the identity function, defined by f(x) = x. x y x
f(x) = x –2 –1 1 2
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16. Basic Polynomial Functions
The squaring function, is defined by f(x) = x2. x y x
f(x) = x2 –2 –1 1 2 4
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17. Basic Polynomial Functions
The cubing function, is defined by f(x) = x3. x y x
f(x) = x3 –2 –1 1 2 –8 8
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18. Graphing Variations of the Identity Function
6.3 Polynomial Functions
EXAMPLE 5
Graphing Variations of the Identity Function
Graph the function by creating a table of ordered pairs. Give the domain and
the range of the function by observing the graph. x y
(a) f(x) = –2x.
Range x
f(x) = –2x –2 4 –1 2
Domain 1 –2 2 –4
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19. Graphing Variations of the Identity Function
6.3 Polynomial Functions
EXAMPLE 5
Graphing Variations of the Identity Function
Graph the function by creating a table of ordered pairs. Give the domain and
the range of the function by observing the graph. x y
(b) f(x) = x2 – 2.
Range x
f(x) = x2 – 2 –2 2 –1 –1
Domain –2 1 –1 2 2
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