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Polynomial Functions
Polynomial Functions
Polynomial is an expression that is either a real number, a variable, or a product of
real numbers and variables with whole number exponents.
Standard Form
Example: x4
+ 2x3 – 3x 2+ 5x + 2
When we write a polynomial we follow the convention that says we write the terms
in order of descending exponents, from left to right.
Classifying Polynomials
Polynomials can be classified by their degree or number of terms.
By degree….
Classifying Polynomials
(continued)
Polynomials can be classified by their degree or number of terms.
By number of terms….
Polynomial Functions
View the link below to learn more about polynomials and
their properties.
Polynomial Functions
FTA
FTA is short for the
Fundamental Theorem
of Algebra
The FTA states that the
number of solutions to a
polynomial equation is
equal to the degree of
the polynomial.
Examples
1) 4x + 2
Degree = 1 (highest
exponent) so the number of
solutions is 1.
2) x2 + 3x + 2
Degree = 2 so the number of
solutions is 2.
3) 4x3 + 3x2 + 2x + 1
Degree = 3 so the number of
solutions is 3.
Zeros or Roots
of a Function
If a polynomial is in factored form, you can use the zero
product property to find values that will make the
polynomial equal zero!
These values are called roots or zeros of the function…also
known as the x-intercepts of the graph.
Example 1
Solve; x2 -4x = 5
Set the equation equal to zero.
Factor the left side of the equation
Use the Zero Product Property
If I multiply the two expressions on the
left and product is equal to zero,
one of the two must be equal to zero.
Set each linear factor equal to zero.
Solve each equation
x2 - 4x – 5 = 0
(x - 5)(x + 1) = 0
(x - 5)= 0
x-5=0
x=5
or (x + 1) = 0
or x + 1 = 0
x = -1
Example 2- 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.
and so we write:
(5 multiplicity 1) and (-1 multiplicity 1)
Example 3- Graphing
Let’s graph x2 – 4x – 5 = 0
First we need to find the vertex.
x = -b/2a
x = -(-4)/2(1) = 4/2 = 2
y = (2)2 – 4(2) – 5 = -9
Vertex = (2, -9)
Then we can graph the
x-intercepts (5, 0) and (-1, 0).
Remember that you can graph
polynomials in the graphing calculator.
Finding Zeros With The Ti-84
Now, let’s use the calculator to find
the zeros of the given polynomial:
f(x) = x2 - 4x - 5
End Behavior
The End Behavior is determined by how the function
“behaves” as you move to the left or right.
Notice as you move along the graph from left to right, the parabola
will fall until you
reach the vertex, and then rise
From the vertex all the way to
Positive infinity.
Let’s Practice
Polynomial
Classification
by Degree
Classification
by Terms
Zeros
Multiplicity of
Zeros
x2 + 7x + 10
4x + 10
3x3 – 12x
Remember that you can graph these polynomials on your
graphing calculator.
End Behavior
Let’s Practice - Solutions
Polynomial
Classification
by Degree
Classification
by Terms
Zeros
Multiplicity
of Zeros
End Behavior
x2 + 7x + 10
2
Trinomial
x=2
x=5
1
1
Left – Rises
Right – Rises
5x + 10
1
Binomial
x=2
1
Left – Falls
Right – Rises
3x3 – 12x
3
Binomial
x=0
x=2
x = -2
1
1
1
Left – Falls
Right – Rises