Basics of a Polynomial

Polynomial

An expression involving a sum of whole number powers multiplied by coefficients:

anxn + … + a2x2 + a1x + a0

Ex: What are examples of polynomials that we have used frequently?

2ax bx c Quadratics:mx bLinear:

Zero or Root of a Polynomial

A value for the independent variable (x) that makes the polynomial equal 0.

Ex: What are the zeros of the following polynomial and how are they represented on the graph?

The zeros/roots are

23 13 10x x

and 523

These are the x-intercepts

Coefficient of a Term

A number that multiplies a variable or variable expression.

Ex: In the polynomial below, what is the coefficient of x4?

7 46 3 13 10x x x

3

Leading Coefficient of a Polynomial

The coefficient of the term in a polynomial which contains the highest power of the variable.

Ex: What is the leading coefficient of the polynomial below and how does it affect the graph? 7 46 3 13 10x x x

6

The leading coefficient acts as the “a” in our polynomial equation (similar to the “a” in ax2+bx+c). It will stretch or

compress the graph and if it is negative, it will flip the graph.

This stretches and flips the graph.

Factor of a Polynomial

The expressions that multiply to get another expression.

Ex: What are the factors of the following polynomial?

Therefore, the factors are

2 3 2x x

and

1 2x x

2x 1x

Example: FactorsWhat do the factors of a polynomial tell us

about the graph of the polynomial?

2 2 2 3y x x x

The values that make each factor equal zero are the x-intercepts.

Example: y-interceptHow can you find the y-intercept of each

equation without a table or graph? 2 2 2 3y x x x 4 3 24 3 10 8y x x x x

In standard form, the term without an “x.”

8

In factored form, the product of the numbers inside of the factors (w/o an

“x”) and the leading coefficient.

2 2 2 324

Remember you can still

substitute 0 for x to find the

y-intercept.

Degree of a Polynomial

Highest power of an independent variable in a polynomial equation.

Ex: What are the degrees of the following polynomials?

74 7 2. 5 5 8 3a x x x

5 2. 7 6 1 3b x x x x 7 6 6 1 3x x x x x

The degree is the number of factors

Example: DegreeWhat does the degree of a polynomial tell us

about the graph of the polynomial? 2 2 2 3y x x x 4 3 24 3 10 8y x x x x

4 221 20y x x x

The degree is the maximum number of roots.Even degrees have the same end behavior.

Degree = 3

Degree = 4

Degree = 4

Odd degrees have opposite end behavior.

Repeated Root of a Polynomial

A value for x that makes more than one factor equal zero.

Ex: What is the repeated root of the polynomial below?

Therefore, the repeated root is

23 2x x 3 3 2x x x

3

Example: Repeated RootsWhat does the degree of a polynomial tell us

about the graph of the polynomial? 30.1 4y x x 22 5y x x

An even repeated root “bounces” off the x-axis.An odd repeated root “twists” through the x-axis.

NOTE: If x is outside of the

parentheses in factored form, 0 is an

x-intercept.

Example: Could Be v Must Be

Odd Repeated Root

“a”:

6

Even

(x + 4), (x – 1), and (x – 5)

-4, 1, and 5

Positive (opens up)

Positive

Minimum Degree:

Degree:

Factors:

x-intercepts:

Orientation:

y-intercept: ~-5

1,2,3 4 5,6Count the

roots

Even Repeated Root

Polynomial Equations to GraphsRoughly Sketch the general shape of:

10 7 12P x x x x

-10 -7 12

Degree = 3

Opposite end behavior (odd)

1 2 3

7 07

xx

12 012

xx

10 010

xx

x-intercepts: Zero-Product Property

Polynomial Equations to GraphsRoughly Sketch the general shape of:

6 3 5 8Q x x x x x

-6 -3 85

5 05

xx

6 06

xx

8 08

xx

x-intercepts: Zero-

Product Property3 0

3xx

Degree = 4

Identical end behavior (even)

1 2 3 4

Degree = 5

Polynomial Equations to GraphsRoughly Sketch the general shape of:

4 2 6 10 15R x x x x x x

-4 -2 106 15

10 010

xx

4 04

xx

15 015

xx

x-intercepts: Zero-

Product Property2 0

2xx

6 06

xx

1 2 3 4 5

Opposite end behavior (odd)

Negative Orientation (start

“up” then go “down”)

–

Polynomial Equations to GraphsRoughly Sketch the general shape of:

2 27 7W x x x

-7 77 0

7xx

x-intercepts: Zero-

Product Property7 07

xx

22

2 Double Roots (bounce off the

x-axis)Degree = 4

Identical end behavior (even)

1,2 3,4

Polynomial Equations to GraphsRoughly Sketch the general shape of:

22 3 5H x x x x

-2 53

2 02

xx

x-intercepts: Zero-

Product Property

3 03

xx

5 05

xx

Degree = 4

3 4

Identical end behavior (even)

1,2

2

2 Double Roots (bounce off the

x-axis)

Negative Orientation (start

“up” then go “down”)

–

Example: Polynomial Equations to Graphs

Without a calculator describe the general shape of: 4 23 2 5 10H x x x x 3

Positive

At most 4 roots.

(0,-10)

Orientation:

End Behavior:x-intercept(s):

y-intercept:

Identical

They can not be determined since it is not in

factored.

The sign of the leading coefficient determines the

orientation.

4

Whether the degree is even or odd determines the end behavior.

(“up” on both ends)

AND The value of the degree determines the

maximum number of roots.

– 10 The value of the constant term determines the

y-intercept.

Example: Equation of a Polynomial to the Graph

2 35 1 3 6f x x x x x

Degree:

x-intercepts:

Orientation:

y-intercept:

2 35 1 3 6

34020

7

-5, -1, 3, 6

Negative

Double Root

Triple RootDifferent end

behavior (odd)

(since the degree is odd, start “up” then go “down”)

– 1

2,34

5,6,7

(Zero Product Property)

2 30 5 0 1 0 3 0 6

Sketch:

Complete GraphWhen a problem says graph an equation or draw a graph:

Plot key points

On graph paper:

Scale your axes appropriately

Plot points accurately

Pay Attention to end behavior