1. Polynomial Functions

2. 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.

3. Explore Polynomials Linear Function Quadratic Function Cubic Function
Quartic Function

4. 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.

5. 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?

6. 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

7. 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

8. 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

9. 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?

10. 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}

11. 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}

12. 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}

13. 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}