Polynomial Functions

Polynomial FunctionsA 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 PolynomialsYou 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:

Can be rewritten as: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.

003007

37

2345

25

xxxxx

xx

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

FTAExamples

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, 1, or 0.

3) 4x3 + 3x2 + 2x + 1

Degree = 3 so the number of solutions is 3, 2, 1 or 0.

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.

***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.x2 - 4x – 5 = 0

Factor the left side of the equation (x - 5)(x + 1) = 0

Use the Zero Product PropertyIf 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.

(x - 5)= 0     or   (x + 1) = 0

Solve each equation for x x - 5 = 0     or   x + 1 = 0      x = 5                  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-1)=0

x=3 x = -2 x = 1

3, multiplicity of 2-2, multiplicity of 11, 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 11, multiplicity of 3

Let’s recap!

Standard form (left to right)

Factored form

823 3 xx )2)(3)(6( xxx

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.