Name ____________________________________Date ______________Block _____
Adv. Math: Chapter 4
Lesson 1: Polynomial Functions
Homework: pg. 210-211: 15-31 all, 32-46 evens, 57-66 all
Objectives:
 Determine roots of polynomial equations
 Apply the Fundamental Theorem of Algebra
Polynomial In One Variable – An expression containing ONE variable
Coefficients represent COMPLEX NUBMERS (real OR imaginary)
Exponents must be NONnegative integers
ex. 𝟐𝟓𝒙𝟑 + 𝟐𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐𝟓
**Complete the following definitions**
Degree (of a polynomial) Leading Coefficient Polynomial Function -
Zeros (of a function ) -
Ex. 1: Consider the polynomial function 𝒇(𝒙) = 𝒙𝟑 − 𝟔𝒙𝟐 + 𝟏𝟎𝒙 − 𝟖 .
pg. 206
a) State the degree and leading coefficient of the polynomial.
b) Determine whether 4 is a zero of f(x).
**On your own . . .
𝒇(𝒙) = 𝟑𝒙𝟒 − 𝒙𝟑 + 𝒙𝟐 + 𝒙 − 𝟏
a) State the degree and leading coefficient of the polynomial.
b) Determine whether -2 is a zero of f(x).
Polynomial Equations
ex.
𝑥 3 − 6𝑥 2 + 10𝑥 − 8 = 𝟎
** From example 1 we discovered that 4 is a zero of the function . . . . that also
means that 4 is a solution for the equation**
Define:
Root Remember . . .
Imaginary Number -
√−𝟏 =
𝒊𝟐 =
Complex Numbers -
Pure Imaginary Numbers -
Fundamental Theorem of Algebra - Every polynomial equation with degree
greater than zero has at least one root in the set of complex numbers.
**The degree of the polynomial determines the number of possible roots in the set
of complex numbers (real & imaginary).
**look over the graphs on page 207 and read
**Graphs of a function with an ODD degree must cross the x-axis at least how many times?
**Graphs of a function with an EVEN degree must cross the x-axis at least how
many times?
**Real roots are located where on the graph???
**Corollary -
if a and
b are roots of the equation, then the equation must be
(x – a)(x – b) = 0
**Use this knowledge to write polynomials when given roots.
Ex. 2:
pg. 208
a) Write a polynomial equation of least degree with roots 2, 4i, and -4i.
b) Does the equation have an odd or even degree?
How many times does the graph of the related function cross the x-axis?
On your own . . .
a) Write a polynomial equation of least degree with roots 2, 3i, and -3i.
b) Does the equation have an odd or even degree?
How many times does the graph of the related function cross the x-axis?
Ex. 3: State the number of complex roots of the equation 𝟗𝒙𝟒 − 𝟑𝟓𝒙𝟐 − 𝟒 = 𝟎.
pg. 208 Then find the roots and graph the related function.
On your own . . .
State the number of complex roots of the equation 𝟑𝟐𝒙𝟑 − 𝟑𝟐𝒙𝟐 + 𝟒𝒙 − 𝟒 = 𝟎.
Then find the roots and graph the related function.
**HINT: factor using groups . . .