Polynomial Functions .

Polynomial Functions

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Polynomial Functions

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

Polynomial Function in

General Form

Degree Name of Function

1 Linear

2 Quadratic

3 Cubic

4 Quarticedxcxbxaxy 234

dcxbxaxy 23

cbxaxy 2

baxy

Explore Polynomials

Linear 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 Coefficient

The 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







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}


-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







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}


-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 Polynomials

EquationEquation

Factored form & Standard Factored form & Standard formform

X-X-InterceptsIntercepts

Sign of Sign of Leading Leading




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}


-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 Polynomials

EquationEquation







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}


-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 Polynomials

EquationEquation







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}


-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 Polynomials

EquationEquation







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}


-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

Polynomial Functions .

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