Polynomial FunctionsPolynomial Function in

General Form

Degree Name of Function

1 Linear

2 Quadratic

3 Cubic

4 Quartic

The largest exponent within the polynomial determines the degree of the polynomial.

edxcxbxaxy 234

dcxbxaxy 23

cbxaxy 2

baxy

Explore PolynomialsLinear Function

Quadratic Function

Cubic Function

Quartic Function

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-60-55-50-45-40-35-30-25-20-15-10-5

510

Leading CoefficientThe leading coefficient is the coefficient of the first term in a polynomial when the terms are written in descending order by degrees.

For example, the quartic function f(x) = -2x4 + x3 – 5x2 – 10 has a leading

coefficient of -2.

Cubic PolynomialsLook at the two graphs and discuss the questions given below.

1. How can you check to see if both graphs are functions?

3. What is the end behaviour for each graph?

4. Which graph do you think has a positive leading coeffient? Why?

5. Which graph do you think has a negative leading coefficient? Why?

2. How many x-intercepts do graphs A & B have?

Graph B

Graph A

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Cubic PolynomialsEquationEquation

Factored form & Factored form & Standard formStandard form

X-InterceptsX-Intercepts Sign of Sign of Leading Leading

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

Factoredy=(x+1)(x+4)(x-2)

Standardy=x3+3x2-6x-8

-4, -1, 2 Positive

As x, y and x-,

y-

Domain

{x| x Є R}

Range

{y| y Є R}

Factoredy=-(x+1)(x+4)(x-2)

Standardy=-x3-3x2+6x+8

-4, -1, 2 Negative

As x, y- and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R}

The following chart shows the properties of the graphs on the left.

-5 -4 -3 -2 -1 1 2 3 4 5

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

-5 -4 -3 -2 -1 1 2 3 4 5

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

Cubic PolynomialsEquationEquation

Factored form & Factored form & Standard formStandard form

X-InterceptsX-Intercepts Sign of Sign of Leading Leading

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

Factoredy=(x+3)2(x-1)

Standardy=x3+5x2+3x-9

-3, 1 Positive

As x, y and x-,

y-

Domain

{x| x Є R}

Range

{y| y Є R}

Factoredy=-(x+3)2(x-1)

Standardy=-x3-5x2-3x+9

-3, 1 Negative

As x, y- and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R}

The following chart shows the properties of the graphs on the left.

-5 -4 -3 -2 -1 1 2 3 4 5

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

-5 -4 -3 -2 -1 1 2 3 4 5

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

Cubic PolynomialsEquationEquation

Factored form & Factored form & Standard formStandard form

X-InterceptsX-Intercepts Sign of Sign of Leading Leading

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

Factoredy=(x-2)3

Standardy=x3-6x2+12x-8

2 Positive

As x, y and x-, y-

Domain

{x| x Є R}

Range

{y| y Є R}

Factoredy=-(x-2)3

Standardy=-x3+6x2-12x+8

2 Negative

As x, y- and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R}

The following chart shows the properties of the graphs on the left.

-5 -4 -3 -2 -1 1 2 3 4 5

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

-5 -4 -3 -2 -1 1 2 3 4 5

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

Quartic PolynomialsLook at the two graphs and discuss the questions given below.

1. How can you check to see if both graphs are functions?

3. What is the end behaviour for each graph?

4. Which graph do you think has a positive leading coeffient? Why?

5. Which graph do you think has a negative leading coefficient? Why?

2. How many x-intercepts do graphs A & B have?

Graph BGraph A

-5 -4 -3 -2 -1 1 2 3 4 5

-14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

12

14

Quartic PolynomialsEquationEquation

Factored form & Standard Factored form & Standard formform

X-X-InterceptsIntercepts

Sign of Sign of Leading Leading

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

Factoredy=(x-3)(x-1)(x+1)(x+2)

Standardy=x4-x3-7x2+x+6

-2,-1,1,3 Positive

As x, y and x-, y

Domain

{x| x Є R}

Range

{y| y Є R,

y ≥ -12.95}

Factoredy=-(x-3)(x-1)(x+1)(x+2)

Standardy=-x4+x3+7x2-x-6

-2,-1,1,3 Negative

As x, y- and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R,

y ≤ 12.95}

The following chart shows the properties of the graphs on the left.

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

12

14

-10 -8 -6 -4 -2 2 4 6 8 10

-14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

Quartic PolynomialsEquationEquation

Factored form & Standard Factored form & Standard formform

X-X-InterceptsIntercepts

Sign of Sign of Leading Leading

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

Factoredy=(x-4)2(x-1)(x+1)

Standardy=x4-8x3+15x2+8x-16

-1,1,4 Positive

As x, y and x-, y

Domain

{x| x Є R}

Range

{y| y Є R,

y ≥ -16.95}

Factoredy=-(x-4)2(x-1)(x+1)

Standardy=-x4+8x3-15x2-8x+16

-1,1,4 Negative

As x, y- and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R,

y ≤ 16.95}

The following chart shows the properties of the graphs on the left.

-5 -4 -3 -2 -1 1 2 3 4 5

-15

-12

-9

-6

-3

3

6

9

12

15

18

-5 -4 -3 -2 -1 1 2 3 4 5

-18

-15

-12

-9

-6

-3

3

6

9

12

15

Quartic PolynomialsEquationEquation

Factored form & Standard Factored form & Standard formform

X-X-InterceptsIntercepts

Sign of Sign of Leading Leading

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

Factoredy=(x+2)3(x-1)

Standardy=x4+5x3+6x2-4x-8

-2,1 Positive

As x, y and x-, y

Domain

{x| x Є R}

Range

{y| y Є R,

y ≥ -8.54}

Factoredy=-(x+2)3(x-1)

Standardy=-x4-5x3-6x2+4x+8

-2,1 Negative

As x, y- and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R,

y ≤ 8.54}

The following chart shows the properties of the graphs on the left.

-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

Quartic PolynomialsEquationEquation

Factored form & Standard Factored form & Standard formform

X-X-InterceptsIntercepts

Sign of Sign of Leading Leading

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

Factoredy=(x-3)4

Standardy=x4-12x3+54x2-108x+81

3 Positive

As x, y and x-, y

Domain

{x| x Є R}

Range

{y| y Є R,

y ≥ 0}

Factoredy=-(x-3)4

Standardy=-x4+12x3-54x2+108x-81

3 Negative

As x, y- and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R,

y ≤ 0}

The following chart shows the properties of the graphs on the left.

-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

-5 -4 -3 -2 -1 1 2 3 4 5

-10

-8

-6

-4

-2

2

4

6

8

10

Multiplicity Let’s look at how we solved for x. (x – 5)(x + 1) = 0

Multiplicity is how often a certain root is part of the factoring. Notice that (x – 5)(x + 1) = 0 only occurred once so the multiplicity for (x – 5) and (x + 1) is 1.