basics of a polynomial. polynomial an expression involving a sum of whole number powers multiplied...
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Basics of a Polynomial
Polynomial
An expression involving a sum of whole number powers multiplied by coefficients:
anxn + … + a2x2 + a1x + a0
Ex: What are examples of polynomials that we have used frequently?
2ax bx c Quadratics:mx bLinear:
Zero or Root of a Polynomial
A value for the independent variable (x) that makes the polynomial equal 0.
Ex: What are the zeros of the following polynomial and how are they represented on the graph?
The zeros/roots are
23 13 10x x
and 523
These are the x-intercepts
Coefficient of a Term
A number that multiplies a variable or variable expression.
Ex: In the polynomial below, what is the coefficient of x4?
7 46 3 13 10x x x
3
Leading Coefficient of a Polynomial
The coefficient of the term in a polynomial which contains the highest power of the variable.
Ex: What is the leading coefficient of the polynomial below and how does it affect the graph? 7 46 3 13 10x x x
6
The leading coefficient acts as the “a” in our polynomial equation (similar to the “a” in ax2+bx+c). It will stretch or
compress the graph and if it is negative, it will flip the graph.
This stretches and flips the graph.
Factor of a Polynomial
The expressions that multiply to get another expression.
Ex: What are the factors of the following polynomial?
Therefore, the factors are
2 3 2x x
and
1 2x x
2x 1x
Example: Factors
What do the factors of a polynomial tell us about the graph of the polynomial?
2 2 2 3y x x x
The values that make each factor equal zero are the x-intercepts.
Example: y-intercept
How can you find the y-intercept of each equation without a table or graph?
2 2 2 3y x x x 4 3 24 3 10 8y x x x x
In standard form, the term without an “x.”
8
In factored form, the product of the numbers inside of the factors (w/o an
“x”) and the leading coefficient.
2 2 2 3
24
Remember you can still
substitute 0 for x to find the
y-intercept.
Degree of a Polynomial
Highest power of an independent variable in a polynomial equation.
Ex: What are the degrees of the following polynomials?
74 7 2. 5 5 8 3a x x x
5 2. 7 6 1 3b x x x x
7 6 6 1 3x x x x x The degree is the number of factors
Example: Degree
What does the degree of a polynomial tell us about the graph of the polynomial? 2 2 2 3y x x x 4 3 24 3 10 8y x x x x
4 221 20y x x x
The degree is the maximum number of roots.Even degrees have the same end behavior.
Degree = 3
Degree = 4
Degree = 4
Odd degrees have opposite end behavior.
Repeated Root of a Polynomial
A value for x that makes more than one factor equal zero.
Ex: What is the repeated root of the polynomial below?
Therefore, the repeated root is
23 2x x 3 3 2x x x
3
Example: Repeated Roots
What does the degree of a polynomial tell us about the graph of the polynomial?
30.1 4y x x 22 5y x x
An even repeated root “bounces” off the x-axis.
An odd repeated root “twists” through the x-axis.
NOTE: If x is outside of the
parentheses in factored form, 0 is an
x-intercept.
Example: Could Be v Must Be
Odd Repeated Root
“a”:
6
Even
(x + 4), (x – 1), and (x – 5)
-4, 1, and 5
Positive (opens up)
Positive
Minimum Degree:
Degree:
Factors:
x-intercepts:
Orientation:
y-intercept: ~-5
1,2,3 4 5,6Count the
roots
Even Repeated Root
Polynomial Equations to GraphsRoughly Sketch the general shape of:
10 7 12P x x x x
-10 -7 12
Degree = 3
Opposite end behavior (odd)
1 2 3
7 0
7
x
x
12 0
12
x
x
10 0
10
x
x
x-intercepts: Zero-Product Property
Polynomial Equations to GraphsRoughly Sketch the general shape of:
6 3 5 8Q x x x x x
-6 -3 85
5 0
5
x
x
6 0
6
x
x
8 0
8
x
x
x-intercepts: Zero-
Product Property
3 0
3
x
x
Degree = 4
Identical end behavior (even)
1 2 3 4
Degree = 5
Polynomial Equations to GraphsRoughly Sketch the general shape of:
4 2 6 10 15R x x x x x x
-4 -2 106 15
10 0
10
x
x
4 0
4
x
x
15 0
15
x
x
x-intercepts: Zero-
Product Property
2 0
2
x
x
6 0
6
x
x
1 2 3 4 5
Opposite end behavior (odd)
Negative Orientation (start
“up” then go “down”)
–
Polynomial Equations to GraphsRoughly Sketch the general shape of:
2 27 7W x x x
-7 77 0
7
x
x
x-intercepts: Zero-
Product Property
7 0
7
x
x
22
2 Double Roots (bounce off the
x-axis)Degree = 4
Identical end behavior (even)
1,2 3,4
Polynomial Equations to GraphsRoughly Sketch the general shape of:
22 3 5H x x x x
-2 53
2 0
2
x
x
x-intercepts: Zero-
Product Property
3 0
3
x
x
5 0
5
x
x
Degree = 4
3 4
Identical end behavior (even)
1,2
2
2 Double Roots (bounce off the
x-axis)
Negative Orientation (start
“up” then go “down”)
–
Example: Polynomial Equations to Graphs
Without a calculator describe the general shape of:
4 23 2 5 10H x x x x 3
Positive
At most 4 roots.
(0,-10)
Orientation:
End Behavior:
x-intercept(s):
y-intercept:
Identical
They can not be determined since it is not in
factored.
The sign of the leading coefficient determines the
orientation.
4
Whether the degree is even or odd determines the end behavior.
(“up” on both ends)
AND The value of the degree determines the
maximum number of roots.
– 10 The value of the constant term determines the
y-intercept.
Example: Equation of a Polynomial to the Graph
2 35 1 3 6f x x x x x
Degree:
x-intercepts:
Orientation:
y-intercept:
2 35 1 3 6
34020
7
-5, -1, 3, 6
Negative
Double Root
Triple RootDifferent end
behavior (odd)
(since the degree is odd, start “up” then go “down”)
– 1
2,34
5,6,7
(Zero Product Property)
2 30 5 0 1 0 3 0 6
Sketch:
Complete Graph
When a problem says graph an equation or draw a graph:
Plot key points
On graph paper:
Scale your axes appropriately
Plot points accurately
Pay Attention to end behavior