# Polynomial Functions

Definition of a Polynomial Function

Let n be a nonnegative integer and let
be real numbers, with
.

The function defined by
is called a polynomial function of x

of degree n. The number a_{n} , the coefficient of the variable to the highest
power, is called the

**leading coefficient**.

**Note:** The variable is only raised to positive integer powers–no negative
or fractional exponents.

However, the coefficients may be any real numbers, including fractions or
irrational numbers

like π or
.

Graph Properties of Polynomial Functions

Let P be any nth degree polynomial function with real coefficients.

**The graph of P has the following properties.**

1. P is continuous for all real numbers, so there are no breaks, holes, jumps in
the graph.

2. The graph of P is a smooth curve with rounded corners and no sharp corners.

3. The graph of P has at most n x-intercepts.

4. The graph of P has at most n – 1 turning points.

**Example 1:** Given the following polynomial functions, state the leading
term, the degree of the

polynomial and the leading coefficient.

a. P(x) = 7x^{4} - 5x^{3} + x^{2} - 7x + 6

b. P(x) = (3x + 2)(x - 7)^{2} (x + 2)^{3}

End Behavior of a Polynomial

Odd-degree polynomials look like y = ±x^{3} .

Even-degree polynomials look like y = ±x^{2} .

Power functions:

A power function is a polynomial that takes the form f (x) = ax^{n} ,
where n is a positive integer.

Modifications of power functions can be graphed using transformations.

Even-degree power functions: f (x) = x ^{4} |
Odd-degree power functions: f (x) = x ^{5} |

**Note:** Multiplying any function by a will multiply
all the y-values by a. The general shape will

stay the same. Exactly the same as it was in section 3.4.

**Zeros of a Polynomial
**

If f is a polynomial and c is a real number for which f (c) = 0, then c is called a zero of f, or a

root of f.

If c is a zero of f, then

· c is an x-intercept of the graph of f.

· (x - c) is a factor of f.

So if we have a polynomial in factored form, we know all of its x-intercepts.

· every factor gives us an x-intercept.

· every x-intercept gives us a factor.

**Example 1:**

Find the zeros of the polynomial:

P(x) = x

^{3}- 5x

^{2}+ 6x

**Example 2:**Consider the function f (x) = -3x(x - 3)

^{4}(5x - 2)(2x -1)

^{3}(4 - x)

^{2}.

Zeros (x-intercepts):

To get the degree, add the multiplicities of all the factors:

The leading term is:

**Behavior at Intercepts:**

Near an x-intercept, c, the shape of the function is determined by the factor
(x-c) and the power

to which it is raised.

Let’s look again at the graph on the first page

Notice the shape of the graph as it crosses the x-axis at
each intercept.

Definition: The multiplicity of a factor in a polynomial function is the power
to which it is

raised in the fully factored form of the polynomial. The multiplicity of the
factor (x-c) with

no higher power is 1.

Compare the multiplicities of the factors in the above polynomial with the shape
at each

corresponding intercept.

The graph of P(x) = (x - c)^{k} looks like the function f (x) = x^{k}
near the x-intercept c. Since we

know the shape of a power function, we have the following rules:

· Even multiplicity: touches x-axis, but doesn’t cross (looks like a parabola
there).

· Odd multiplicity of 1: crosses the x-axis (looks like a line there).

· Odd multiplicity ≥3 : crosses the x-axis and looks like a cubic there.

**Steps to graphing other polynomials:**

1. Find the y-intercept by finding P(0).

2. Factor and find x-intercepts.

3. Mark x-intercepts on x-axis.

4. For each x-intercept, determine the behavior.

· Even multiplicity: touches x-axis, but doesn’t cross (looks like a parabola
there).

· Odd multiplicity of 1: crosses the x-axis (looks like a line there).

· Odd multiplicity ≥3 : crosses the x-axis and looks like a cubic there.

**Note:** It helps to make a table as shown in the examples below.

5. Determine the leading term.

· Degree: is it odd or even?

· Sign: is the coefficient positive or negative?

6. Determine the end behavior. What does it “look like”?

7. Draw the graph, being careful to make a nice smooth
curve with no sharp corners.

**Note:** without calculus or plotting lots of points, we don’t have enough
information to know how

high or how low the turning points are.

**Example 3:**

Find the zeros then graph the polynomial. Be sure to label the x intercepts, y
intercept if

possible and have correct end behavior.

**Example 4:**

Find the zeros then graph the polynomial. Be sure to label the x intercepts, y
intercept if

possible and have correct end behavior.

**Example 5:**

Find the zeros then graph the polynomial. Be sure to label the x intercepts, y
intercept if

possible and have correct end behavior.

**Example 6:**

Find the zeros then graph the polynomial. Be sure to label the x intercepts, y
intercept if

possible and have correct end behavior.

**Example 7:**

Given the graph of a polynomial determine what the equation of that polynomial.

**Example 8:**

Given the graph of a polynomial determine what the equation of that polynomial.