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Chapter 7:
Polynomial Functions
7.1 Polynomial Functions
Objectives:
1)
Evaluate polynomial functions
2)Identify general shapes of graphs of polynomial functions
Polynomial Functions
Recall that a polynomial is a monomial or sum of monomials.
The expression 3x2 – 3x + 1 is a polynomial in one variable since
it only contains one variable, x.
A polynomial of degree n in one variable x is an expression of the
form a0xn + a1xn-1 + … + an-1x + an where the coefficients a0, a1,
a2…an represent real numbers, a0 is not zero, and n represents a
nonnegative integer.
Examples: 3x5 + 2x4 - 5x3 + x2 + 1
n=5, a0=3, a1=2, a2= -5, a3=1, a4=0, and a5=1
Vocabulary
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
Expression
Degree
Leading Coefficient
Constant
9
0
9
Linear
x-2
1
1
Quadratic
3x2 + 4x - 5
2
3
Cubic
4x3 - 6
3
4
General
a0xn + a1xn-1 + … + an-1x + an
n
a0
Example 1
State the degree and leading coefficient of each polynomial in one
variable. If it is not a polynomial in one variable, explain why.
a) 7x4 + 5x2 + x – 9
b) 8x2 + 3xy – 2y2
c)
d)
7x6
–
4x3
1
+
x
1 2
x + 2x3 – x5
2
Polynomial Function
A polynomial equation used to represent a function.
Examples:
f(x) = 4x2 – 5x + 2 is a quadratic polynomial function.
f(x) = 2x3 + 4x2 – 5x + 7 is a cubic polynomial function.
If you know an element in the domain of any polynomial
function, you can find the corresponding value in the range.
Recall that f(3) can be found by evaluating the function for x = 3.
Example 2
a) Show that the polynomial function f(r)=3r2 – 3r + 1 gives the total
number of hexagons when r = 1, 2, and 3.
b) Find the total number of hexagons in a honeycomb with 12 rings.
Example 3
Functional Values of Variables
Find p(a2) if p(x) = x3 + 4x2 – 5x
Find q(a +1) – 2q(a) if q(x) = x2 + 3x + 4
Graphs of Polynomial Functions
These graphs show the MAXIMUM number of times the graph of each type of polynomial
may intersect the x-axis.
The x-coordinate of the point at which the graph intersects the x-axis is called a zero of a
function.
How does the degree compare to the maximum number of real zeros?
Notice the shapes of the graphs for even-degree polynomial functions and odddegree polynomial functions. The degree and leading coefficient of a polynomial
function determine the graph’s end behavior.
Partner Practice
Text p. 350 #s 1-11 all
Homework
Text p. 350-351 #s 16-38 even
End Behavior
Is the behavior of a graph as x approaches positive infinity (+
) or negative infinity ( ).
This is represented as x +
and x
-
Note on x-intercepts:
Graph of an even-degree function:
-
may or may not intersect the x-axis
-
if it intersects the x-axis in two places, the function has
two real zeros.
-
if it does not intersect the x-axis the roots of the related
equation are imaginary and cannot be determined from
the graph
-
If the graph is tangent to the x-axis, as shown above, there
are two zeros that are the same number
Note on x-intercepts (ct’d):
Graph of an odd-degree function:
- ALWAYS intersect the x-axis at least once
- Thus, ALWAYS has at least one real zero.
Example 4a
1) describe end behavior
2) determine whether it represents an
odd-degree or even-degree polynomial
function
3) state the number of real zeros
Example 4b
1) describe end behavior
2) determine whether it represents
an odd-degree or even-degree
polynomial function
3) state the number of real zeros
Example 4c
1) describe end behavior
2) determine whether it
represents an odd-degree or
even-degree polynomial
function
3) state the number of real zeros
Homework
Text p. 350 #s 12-15 all
Text p. 351 #s 39-48 all