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
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 Polynomials
EquationEquation
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 Polynomials
EquationEquation
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 Polynomials
EquationEquation
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 Polynomials
EquationEquation
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
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