Polynomial Functions
Unit 4
Polynomials
• Monomial—a number, variable, or product of
numbers and variables all raised to whole number
powers
• Polynomial Expression--a monomial or sum of
monomials.
• Polynomial Function—function that is defined by a
polynomial expression
• Leading coefficient—coefficient of highest powered
term
• Degree of polynomial—power of highest powered
variable
Standard Form
The standard form of a polynomial function arranges
the terms by degree in descending numerical order
A polynomial function P(x) in standard form is:
𝑃 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 +. . . +𝑎1 𝑥 + 𝑎0
Where n is a nonnegative integer and 𝑎𝑛 , . . . 𝑎0 are
real numbers.
Ex: 𝑃 𝑥 = 4𝑥 3 + 3𝑥 2 + 5𝑥 − 2
Classifying Polynomials by degree
• d=0 Constant
52
• d=1 Linear
x2
• d=2 Quadratic
x  2x 1
• d=3 Cubic
4x  6
• d=4 Quartic
9 x  8x  2 x  x  2
• d=5 Quintic
 x  3x  2 x
2
3
4
5
3
4
2
Classifying by Number of Terms
• Monomial—one term
• Binomial—two terms
• Trinomial—three terms
• N-nomial—n terms
Classify by degree and number of
terms.
1. 3𝑥 2 + 2𝑥 − 3
2. 4𝑥 5 − 3𝑥
3. 2𝑥 2 − 4𝑥 4 + 6𝑥
4. 4 𝑥 7
Polynomial Function
• Polynomial equation used to represent a
function
f ( x)  4 x 2  5 x  2
P( x)  10 x 3  2 x 2  x  3
Graphs of Polynomial Functions
• Constant
Linear
Quadratic
• Show graphs with positive and negative LC
End Behavior of Graphs
As 𝑥 → ∞, 𝑓(𝑥) → _________
As 𝑥 → −∞, 𝑓(𝑥) → _________
As x gets bigger or smaller, what happens to the
function value?
Graphs of Polynomial Functions
• Quadratic: 𝑓 𝑥 = 𝑥 2 − 2𝑥
D:________
R: ________
Zeros: _______
Inc: _______
Dec: _______
as 𝑥 → ∞, 𝑓(𝑥) → _________
as 𝑥 → −∞, 𝑓(𝑥) → _________
Graphs of Polynomial Functions
• Cubic: 𝑓 𝑥 = 𝑥 3 + 𝑥 2
D:________
R: ________
Zeros: _______
Inc: _______
Dec: _______
as 𝑥 → ∞, 𝑓(𝑥) → _________
as 𝑥 → −∞, 𝑓(𝑥) → _________
Graphs of Polynomial Functions
• Quartic: 𝑓 𝑥 = 4𝑥 4 − 7𝑥 2 − 2
D:________
R: ________
Zeros: _______
Inc: _______
Dec: _______
as 𝑥 → ∞, 𝑓(𝑥) → _________
as 𝑥 → −∞, 𝑓(𝑥) → _________
Graphs of Polynomial Functions
• Quintic: 𝑓 𝑥 = 𝑥 5 − 4𝑥 3
D:________
R: ________
Zeros: _______
Inc: _______
Dec: _______
as 𝑥 → ∞, 𝑓(𝑥) → _________
as 𝑥 → −∞, 𝑓(𝑥) → _________
End Behavior of Graphs
+Lead Coefficient
Even Degree
Odd Degree
-Lead Coefficient
End Behavior of Graphs
As 𝑥 → ∞, 𝑓(𝑥) → _________
As 𝑥 → −∞, 𝑓(𝑥) → _________
End Behavior of Graphs
As 𝑥 → ∞, 𝑓(𝑥) → _________
As 𝑥 → −∞, 𝑓(𝑥) → _________
To sketch the graph
• Determine the end
behavior.
• Determine the x
intercepts (where y=0)
f ( x)   x 2  2 x
0  x2  2x
0   x( x  2)
 x  0 and x  2  0
x  0 and x  2
Turning Points
A polynomial function of degree n has at
most n-1 turning points.
1. 𝑓 𝑥 = 3𝑥 5 − 4𝑥 3 + 3𝑥 2 + 2
2. 𝑓 𝑥 = 𝑥 3 − 3𝑥 4 + 2𝑥 2 − 1
3. 𝑦 = 𝑥 2 + 2𝑥 + 3