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![Page 1: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/1.jpg)
![Page 2: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/2.jpg)
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
![Page 3: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/3.jpg)
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
![Page 4: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/4.jpg)
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.
![Page 5: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/5.jpg)
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
![Page 6: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/6.jpg)
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
![Page 7: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/7.jpg)
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
![Page 8: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/8.jpg)
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
![Page 9: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/9.jpg)
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
![Page 10: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/10.jpg)
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
![Page 11: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/11.jpg)
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
![Page 12: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/12.jpg)
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
![Page 13: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/13.jpg)
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
![Page 14: Polynomial Functions Polynomial Function in General Form DegreeName of Function 1Linear 2Quadratic 3Cubic 4Quartic The largest exponent within the polynomial.](https://reader035.fdocuments.net/reader035/viewer/2022062719/56649ed45503460f94be558a/html5/thumbnails/14.jpg)
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.