Algebra 2 A
Section 5.3 Notes: Polynomial Functions
A polynomial is an expression that is a sum of variables and exponents.
The coefficient of the first term of a polynomial in standard form is called the leading coefficient.
Degree
0
1
2
3
4
General
Type
Constant
Linear
Quadratic
Cubic
Quartic
Degree n
Expression
12
4x – 9
5x2 – 6x – 9
8x3 + 12x2 – 3x + 1
x2 – 4x4 + 3x
an x n  an 1 x n 1  ...  a1 x  a0
Leading coefficient
12
4
5
8
–4
an
**You cannot have a polynomial with two different variables.
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) 7z3 – 4z2 + z
b) 6a3 – 4a2 + ab2
c) 3x5 + 2x2 – 4 – 8x6
A polynomial function is a continuous function that can be described by a polynomial equation in one variable. The simplest
polynomial functions of the form f(x) = axb where a and bare nonzero real numbers are called power functions.
Example 2: The volume of air in the lungs during a 5-second respiratory cycle can be modeled by v(t) = –0.037t 3 + 0.152t 2 + 0.173t,
where v is the volume in liters and t is the time in seconds. This model is an example of a polynomial function. Find the volume of air
in the lungs 1.5 seconds into the respiratory cycle.
Example 3: Find p(–4) and p(2) for each function.
a) p(x) = –x3 + 3x2 – 5
b) p(x) = x4 – 2x2 + 3
The general shapes of the graphs of several polynomial functions show the maximum number of times the graph of each function may
intersect the x axis. This is the same number as the degree of the polynomial.
The domain of any polynomial function is all real numbers. The end behavior is the behavior of the graph of f(x) as x approaches
positive infinity (x → +∞) or negative infinity (x → –∞). The degree and leading coefficient of a polynomial function determine the
end behavior of the graph and the range of the function.
The number of real zeros of a polynomial function can be determined by examining its graph. Recall that real zeros occur at xintercepts, so the number of times a graph crosses the x-axis equals the number of real zeros.
Example 1: For each graph,
*Describe the end behavior
*Determine whether it represents an odd-degree or an even-degree polynomial function
*State the number of real zeros
a)
b)
c)