Transcript Module 4 Lesson 4 Basics of Polynomials Notes

Polynomial Functions
Polynomial Functions
A 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 of a Polynomial Expression
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.
Polynomials can be classified by their degree or number of terms.
The degree can tell us how many possible solutions a polynomial can have.
Classifying Polynomials
You may or may not see all terms included in the written form of a polynomial. If
a term is missing then the term can be written in if needed but you must give it
a coefficient of 0. For example:
7 x 5  3x 2
Can be rewritten as:
7 x 5  0 x 4  0 x 3  3x 2  0 x  0
and this does not change the degree or classification of the polynomial.
Here are a few examples of the polynomials showing their classifications and
types.
Polynomial
Degree
Classification
Type
-7
0
Constant
Monomial
6x
1
Linear
Monomial
2x2 – 6x + 8
2
Quadratic
Trinomial
x3 + 7x - 2
3
Cubic
Trinomial
x4 + 7x2
4
Quartic (or 4th degree)
Binomial
-x6 +4x5 – 3x3 – 2x -1
6
6th degree
Polynomial
Examples
FTA
1) 4x + 2
FTA is short for the
Fundamental Theorem
of Algebra
The FTA states that the
maximum number of
possible solutions to a
polynomial equation is
equal to the degree of
the polynomial.
 Degree = 1 (highest
exponent) so the number of
solutions is 1.
2) x2 + 3x + 2
 Degree = 2 so the number of
solutions is 2, 1, or 0.
3) 4x3 + 3x2 + 2x + 1
 Degree = 3 so the number of
solutions is 3, 2, 1 or 0.
***Number of solutions = # of times the graph crossed the x-axis
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 – or in other words, find the
solution(s)!
These values are called roots or zeros of the
function…these points are also known as the
x-intercepts or solutions 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 for x
x2 - 4x – 5 = 0
(x - 5)(x + 1) = 0
(x - 5)= 0
or (x + 1) = 0
x-5=0
x=5
or x + 1 = 0
x = -1
The zeros, or roots, are x = -1, 5
So, making these x values into ordered pairs gives us
solutions or x-intercepts of (0, -1) and (0, 5). This is
where the graph crosses the x-axis.
Multiplicity using Example 1
Let’s look at how we solved for x in example 1. In this
example we only had two binomials, x-5 and x+1.
(x – 5)(x + 1) = 0
Multiplicity is how often a certain factor appears in the
polynomial.
Notice that (x – 5)(x + 1) = 0 only occurred once so the
multiplicity for (x – 5) and (x + 1) is 1.
Multiplicity extended
If we have a polynomial that has a higher degree we can have
more than 2 solutions.
For example, X6 – 7x5 + 12x4 + 14x3 – 59x2 + 57x - 18
If we factored this we would get (x-3)(x-3)(x+2)(x-1)(x-1)(x-1)
Which could be rewritten as
(x-3) 2(x+2)(x-1) 3
Notice, either way it is easy to see what binomials repeat and
how many times, this is multiplicity.
Once we solve the binomials for x, find the zeros, we can write
the multiplicity statement:
(x-3) = 0 (x+2)=0
x=3
x = -2
(x-1)=0
x=1
3, multiplicity of 2
-2, multiplicity of 1
1, multiplicity of 3
Multiplicity
If you graphed the equation on the last slide you may still be
saying to yourself “The graph appears to only cross the x-axis
three times. What is this multiplicity all about? What does it
look like?”
(x-3) 2(x+2)(x-1) 3
This polynomial still only has three (3) zeros, but we say it has
six (6) zeros counting multiplicity since multiplicity just tells
us the number of times the FACTOR appears in the
polynomial.
3, multiplicity of 2
-2, multiplicity of 1
1, multiplicity of 3
Let’s recap!
Standard form (left to right)
 3x 3  2 x  8
Factored form
( x  6)( x  3)( x  2)
The FTA (Fundamental Theorem of Algebra) states that the maximum number
of possible solutions to a polynomial equation is equal to the degree of the
polynomial.
If a polynomial is in factored form, you can use the zero product property to
find values that will make the polynomial equal zero – or in other words, find
the solution(s)!
Zeros, roots, solutions and
x-intercepts are all closely
related. They can be written as
x = #, #, # etc. But we can
rewrite as ordered pairs.
Multiplicity refers to how
many times a factor appears in
the factored form of the
polynomial.