Ch.2 Polynomial Functions
Lesson 2.1 Polynomials
Objective: Students will be able to identify a polynomial function, to evaluate it
using synthetic substitution and to determine its zeros.
Polynomial: can be written in the form…..
Standard form: when the powers of x are written in descending order.
Leading term:
leading coefficient:
Name
Degree
Constant
Linear
Quadratic
Cubic
Quartic
Quintic
degree:
Example
Example: State whether the following are polynomial functions. Give the zeros of
the function, if they exist.
a. F(x) = 2x3 – 32x
b. 𝑓(𝑥) =
𝑥+1
𝑥−1
Example: If P(x) = 3x4-7x3-5x2+9x+10, evaluate the following.
a. P(2)
b. P(-3n)
Synthetic Substitution: another way to evaluate polynomials
Example:
a. P(x) = 3x4-7x3 – 5x2 + 9x + 10, find P(2)
b. S(x) = 3x4-5x2+9x+1-, find S(-2)
Synthetic Division: Used to divide a polynomial by a binomial of the form x-a
Divide 2x4 – 15x2 -10x + 5 by x-3
Examples: Use synthetic division to divide the polynomial by the linear factor.
1. (3x2+ 7x + 2) ÷ (x + 2)
2. (2x2+ 7x – 15) ÷ (x + 5)
3. (7x2– 3x + 5) ÷ (x + 1)
4. (4x2+ x + 1) ÷ (x – 2)
5. (3x2+ 4x – x4– 2x3– 4) ÷ (x + 2)
6. (3x2– 4 + x3) ÷ (x – 1)
HW: p. #2-10 even, 16, 19, 20
Lesson 2.2 Synthetic Division, The Remainder and Factor Theorems
Objective: Students will be able to use synthetic division and to apply the
remainder and factor theorems.
DO NOW:
Divide the following using synthetic division.
a. X3 – 2x2 + 5x + 1; x-1
b. x4 – 2x3 + 5x + 2; x-1
Use Synthetic Substitution to find P(3) if P(x) = 3x3 – 4x2 + 2x -5
THE REMAINDER THEOREM: When a polynomial P(x) is divided by x-a, the
remainder is P(a).
Ex. Find the remainder when P(x) is divided by the given divisor.
P(x) = 5x3 – 4x2 – 2x + 1
a. P(1)
b. P(4)
THE FACTOR THEOREM: For a polynomial P(x), x-a is a factor if and only if P(a) = 0.
Ex. If P(x) = 2x4 + 5x3 – 8x2 = 17x -6, determine whether each of the following is a
factor of P(x):
a. x-1
b. x-2
Ex. Is x+1 a factor of P(x) = x3 + 3x2 + x -1? If so, find the other factors of P(x).
Ex. You are given a polynomial equation and one or more of its roots. Find the
remaining roots.
a. 2x3 – 5x2 – 4x + 3; root: x=3
b. b. 2x4-9x3+2x2 + 9x – 4; roots: x=-1, x=1
HW: p. 61 # 12-24 even
Graphs of Polynomial Functions
Objective: Students will be able to describe end behavior using limit notation, find
the zeros of a function and use multiplicity rules to graph.
DO NOW: Graph the following parent functions. Make a table or use your
graphing calculator
y=x2
y=x
y=x3
y = x4
Graph Properties of Polynomial Functions
Let P be any nth degree polynomial function with real coefficients.
The graph of P has the following properties.
1. P is continuous for all real numbers, so there are no breaks,
holes, jumps in the graph.
2. The graph of P is a smooth curve with rounded corners and no
sharp corners.
3. The graph of P has at most n x-intercepts.
4. The graph of P has at most n – 1 turning points.
 End behavior of polynomials:
What happens to the graph as x approaches ±∞
*The end behavior of a polynomial is closely related to the end
behavior of its leading term. *
Example: Comparing the Graphs of a Polynomial and its Leading Term
Graph f(x) = x3 and g(x) = x3 - 4x2 - 5x - 3 in the calculator.
Continue zooming out. What happens?
In general….
Leading Term Test for Polynomial End Behavior
For any polynomial function, the limits as x goes to positive and
negative infinity are determined by the degree n of the polynomial and
its leading coefficient An.
An> 0 and n is odd
An< 0 and n is odd
An> 0 and n is even
An<0 and n is even
Example 1: Given the following polynomial functions, state the degree
of the function and describe the end behavior using limit notation.
a.
P( x )  7 x 4  5 x 3  x 2  7 x  6
b.
P( x )  (3x  2)( x  7)2 ( x  2)3
Example: Find the zeros of the polynomial and use the end behavior to
draw a rough sketch of the graph.
P( x )  x 3  5 x 2  6 x
f ( x )  x( x  2)3 ( x  1)2
 Even multiplicity: touches x-axis, but doesn’t cross (looks like a
parabola there).
 Odd multiplicity of 1: crosses the x-axis (looks like a line there).
 Odd multiplicity >3
there.
: crosses the x-axis and looks like a cubic
Example: Determine the end behavior, zeros, and multiplicity of each zero for the following. Then graph.
P(x) = (x+2)3(x-1)2
Lesson 2.4 Rational Root Theorem
Objective: Students will be able to apply the rational root theorem to test roots in
order to determine the zeros of a function.
DO NOW: Find the roots of the following polynomials.
a. f(x) = 15x3 + 5x2 + 3x + 1
b. f(x) = 4n3 – 12n2 + 3n -9
Let’s look at the calculator to determine the zeros of
P( x )  x 3  5 x 2  6 x
Now, let’s look at the calculator to determine the zeros of
P(x) = 2x4 + x3 – 31x2 – 26x + 24
The Rational Zero Theorem:
WHY IS IT IMPORTANT: Narrows the search for rational zeros to a finite list.
 If P( x)  a n x n  a n 1 x n1    a1 x  a0 has integer coefficients a n  0
and
p
is a rational zero (in lowest terms) of p, then
q
p is a factor of the constant term a 0 and q is a factor of the leading
coefficient a n .
Use the Rational Zeros Theorem to list all the Possible Rational Zeros of the
following polynomials
EX1: f(x) = X3 – 4x2 + 15
EX2:
f (x) = 10x3 + 7x2-2x-6
EX3: Find the roots of f(x) = x3 – 2x2 -11x +12
HINT: Apply the Rational Root Theorem to find the possible rational roots!
What is p? ________
What is q? ________
EX4: Find the roots of f(x) = 3x3 – x2 -15x + 5
EX 5] Find the possible rational roots of 3x3  5x2  7 x  2  0 .
Lesson 2.5 Conjugate Pairs
Objective: Students will be able to find conjugate pairs of complex zeros. They
will also be able to write a polynomial function given the zeros.
Do now:
a. Find the polynomial with integer coefficients having roots at 3, –
5, and –½, and passing through (–1, 16). Write the answer in
standard form.
b. Find the zeros of the given polynomial. P(x) = x4 – 3x3 + 6x2 + 2x – 60
What do you notice about the complex zeros in example b?
Complex Conjugate Zeros Theorem:
If f is a polynomial function and a +bi is a zero of f , then a - bi is also a
zero of f .
Irrational Conjugate Zeros Theorem:
If f is a polynomial function and 2+√3 is a zero of f , then 2-√3 is also a
zero of f .
EX: Find a 3rd degree polynomial in standard form with integer coefficients given
-2, and 3i are zeros.
EX: Find a polynomial with integer coefficients given the zeros at 1 and 2 - 3i.
EX: Write a polynomial with integer coefficients with degree 4 and zeros at -2
(multiplicity 2) and - i .
EX: Write a polynomial with integer coefficients with zeros at 4 and 3 + √2.
EX: Use the given zero to find the remaining zeros of each polynomial function.
P(x) = 2x3 – 5x2+6x -2 Zero: 1+i
HW: Worksheet