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Polynomial Functions

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**Polynomial Function in General Form**

Polynomial Functions Polynomial Function in General Form Degree Name of Function 1 Linear 2 Quadratic 3 Cubic 4 Quartic Teachers: This definition for ‘degree’ has been simplified intentionally to help students understand the concept quickly and easily. The largest exponent within the polynomial determines the degree of the polynomial.

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**Explore Polynomials Linear Function Quadratic Function Cubic Function**

Quartic Function

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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.

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Cubic Polynomials Look at the two graphs and discuss the questions given below. Graph B Graph A 1. How can you check to see if both graphs are functions? 2. How many x-intercepts do graphs A & B have? 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?

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**Factored form & Standard form Sign of Leading Coefficient**

Cubic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x+1)(x+4)(x-2) Standard y=x3+3x2-6x-8 -4, -1, 2 Positive As x, y and x-, y- Domain {x| x Є R} Range {y| y Є R} y=-(x+1)(x+4)(x-2) y=-x3-3x2+6x+8 Negative As x, y- and x-, y

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**Factored form & Standard form Sign of Leading Coefficient**

Cubic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x+3)2(x-1) Standard y=x3+5x2+3x-9 -3, 1 Positive As x, y and x-, y- Domain {x| x Є R} Range {y| y Є R} y=-(x+3)2(x-1) y=-x3-5x2-3x+9 Negative As x, y- and x-, y

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**Factored form & Standard form Sign of Leading Coefficient**

Cubic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x-2)3 Standard y=x3-6x2+12x-8 2 Positive As x, y and x-, y- Domain {x| x Є R} Range {y| y Є R} y=-(x-2)3 y=-x3+6x2-12x+8 Negative As x, y- and x-, y

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Quartic Polynomials Look at the two graphs and discuss the questions given below. Graph B Graph A 1. How can you check to see if both graphs are functions? 2. How many x-intercepts do graphs A & B have? 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?

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**Factored form & Standard form Sign of Leading Coefficient**

Quartic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x-3)(x-1)(x+1)(x+2) Standard y=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 ≥ } y=-(x-3)(x-1)(x+1)(x+2) y=-x4+x3+7x2-x-6 Negative As x, y- and x-, y- y ≤ 12.95}

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**Factored form & Standard form Sign of Leading Coefficient**

Quartic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x-4)2(x-1)(x+1) Standard y=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 ≥ } y=-(x-4)2(x-1)(x+1) y=-x4+8x3-15x2-8x+16 Negative As x, y- and x-, y- y ≤ 16.95}

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**Factored form & Standard form Sign of Leading Coefficient**

Quartic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x+2)3(x-1) Standard y=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} y=-(x+2)3(x-1) y=-x4-5x3-6x2+4x+8 Negative As x, y- and x-, y- y ≤ 8.54}

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**Factored form & Standard form Sign of Leading Coefficient**

Quartic Polynomials The following chart shows the properties of the graphs on the left. Equation Factored form & Standard form X-Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x-3)4 Standard y=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} y=-(x-3)4 y=-x4+12x3-54x2+108x-81 Negative As x, y- and x-, y- y ≤ 0}

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Polynomial Functions A polynomial in x is a sum of monomials* in x.

Polynomial Functions A polynomial in x is a sum of monomials* in x.

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