ALGEBRA 2
5.3 Day 1
Polynomial Functions
RECALL
A polynomial is an expression that is a sum of
variables and exponents.
Degree
0
1
2
3
4
5+
Type
Constant
Linear
Quadratic
Cubic
Quartic
Degree n
Example
12
4𝑥 − 9
5𝑥 2 − 6𝑥 − 9
8𝑥 3 + 12𝑥 2 − 3𝑥 + 1
𝑥 2 − 4𝑥 4 + 3𝑥
Examples vary
LEADING COEFFICIENT
The coefficient of the first term of a polynomial in
standard form is called the leading coefficient.
Example
Leading coefficient
12
12
4𝑥 − 9
4
5𝑥 2 − 6𝑥 − 9
5
8𝑥 3 + 12𝑥 2 − 3𝑥 + 1
8
𝑥 2 − 4𝑥 4 + 3𝑥
−4

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) 7𝑧 3 − 4𝑧 2 + 𝑧
b) 6𝑎3 − 4𝑎2 + 𝑎𝑏 2
c) 3𝑥 5 + 2𝑥 2 − 4 − 8𝑥 6
REAL WORLD – EXAMPLE 2
RESPIRATION 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
If 𝑏 𝑚 = 2𝑚2 + 𝑚 − 1, find 𝑏 2𝑥 − 1 − 3𝑏(𝑥).
Step 1: Find 𝑏 2𝑥 − 1

Step 2: Find 3𝑏(𝑥)
Step 3: Now subtract step 1 and 2.
YOU TRY
Find g(2x + 1) – 2g(x)
A.
1
B.
2x 2 + 4x – 2
C.
2x 2 + 4x + 10
D.
2x 2 – 2
if g(b) = b2 + 3.
Hint:
Step 1: Find 𝑔(2𝑥 + 1)
Step 2: Find 2𝑔(𝑥)
Step 3: Now subtract step 1 and 2.
EXIT SLIP
1. Determine whether 3x3 + 2x2 – 3 is a polynomial
in one variable. If so, state the degree and leading
coefficient.
2. Find 𝑓 2𝑥 − 𝑓(𝑥)
if 𝑓 𝑥 = 𝑥 2 + 3𝑥 − 4
ALGEBRA 2
5.3 Day 2
Polynomial Functions
Zeros of Even- and Odd-Degree Functions
 Odd-degree functions will always have an odd
number of real zeros.
 Even-degree functions will always have an even
number of real zeros or no zeros at all.

The number of turns is always one less than the
degree.
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)
YOU TRY
For the graph, determine whether it represents an
odd-degree or an even-degree function, and state
the number of real zeros.
EXIT SLIP
For the graph,
 Describe the end behavior
 Determine whether it
represents an odd-degree or
an even-degree polynomial
function
 State the number of real zeros