Pre-Calculus
Unit 2
Section 2.2 Notes – Polynomial Functions
Objectives
 graph polynomial functions
 model real-world data with polynomial functions
Let n be a nonnegative integer and let a0, a1, a2, …, an – 1, an be real numbers with an ≠ 0. Then the function given by
f(x) = anxn + an – 1 xn – 1 + … + a2x2 + a1x + a0
is called a polynomial function of degree n. The leading coefficient of a polynomial function is the coefficient of the variable with
the greatest exponent. The leading coefficient of f(x) is an.
You are already familiar with the following polynomial functions.
Example 1: Describe the transformation of the graph, then give domain, range, intercepts, end behavior, continuity, & increasing
and/or decreasing intervals.
a) f (x) = (x – 3)5
b) f (x) = x 6 – 1
Example 2: Without a calculator, describe the end behavior of the graph using limits. Explain your reasoning using the leading term
test.
a) f (x) = 3x 4 – x 3 + x 2 + x – 1
b) f (x) = –3x 2 – 2x 5 – x 3
Turning Points: where the graph of a function changes from increasing to decreasing and vice versa.
 How are zeros related to turning points???
Example 3: State the number of possible real zeros and turning points of the given function. Then determine all of the real zeros by
factoring.
a) f (x) = x 3 + 5x 2 + 4x
b) f (x) = x 4 – 13x 2 + 36
Example 4: State the number of possible real zeros and turning points. Then determine all of the real zeros by factoring.
a) g (x) = x 4 – 4x 2 + 3
b) h (x) = x 5 - 5x 3 - 6x
Example 5: State the number of possible real zeros and turning points of h (x) = x 4 + 5x 3 + 6x 2. Then determine all of the real zeros
by factoring.
Example 6: For f (x) = x(3x + 1)(x – 2) 2
a) Apply the leading-term test.
b) Determine the zeros and state the multiplicity of any repeated zeros.
c) Sketch the graph (without a calculator).
Example 6: Choose the zeros and state the multiplicity of any repeated zeros for f (x) = 3x(x + 2)2(2x – 1)3.
a) 0, –2 (multiplicity 2), ½ (multiplicity 3)
b) 2 (multiplicity 2), – ½ (multiplicity 3)
c) 4 (multiplicity 2), 1/8 (multiplicity 3)
d) –2 (multiplicity 2), ½ (multiplicity 3)