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: degree:

Name Degree Example

Constant

Linear

Quadratic

Cubic

Quartic

Quintic

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+1x−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 = x y=x2

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.

b.

Example: Find the zeros of the polynomial and use the end behavior to draw a rough sketch of the graph.

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 : crosses the x-axis and looks like a cubic there.

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

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 ) = an xn + an−1 xn−1 + ⋯ + a1 x + a0 has integer coefficients (an ≠ 0 )

and pq is a rational zero (in lowest terms) of p, then

p is a factor of the constant term a0 and q is a factor of the leading

coefficient an .

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 .

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