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# Polynomial Functions and Their Graphs

Math 1310
Section 4.1: Polynomial Functions and Their Graphs

A polynomial function is a function of the form where are real numbers and n is a whole number.

The degree of the polynomial function is n. We call the term the leading term, and is called the leading coefficient. Our objectives in working with polynomial functions will be, first, to gather information
about the graph of the function and, second, to use that information to generate a reasonably good
graph without plotting a lot of points. In later examples, we’ll use information given to us about
the graph of a function to write its equation.

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  We’ll start with the shapes of the graphs of functions of the form You should be familiar with the graphs of and The graph of is even, will resemble the graph of , and the graph
of n is odd, will resemble the graph of  Next, you will need to be able to describe the end behavior of a function.
End Behavior of Polynomial Functions
The behavior of a graph of a function to the far left or far right is called its end behavior.
The end behavior of a polynomial function is revealed by the leading term of the polynomial
function.
1. Even-degree polynomials look like .  2. Odd-degree polynomials look like.  Next, you should be able to find the x intercept(s) and the y intercept of a polynomial function.

Zeros of Polynomial Functions

You will need to set the function equal to zero and then use the Zero Product Property to find the
x-intercept(s). That means if ab = 0, then either a = 0 or b = 0 . To find the y intercept of a function,
you will find f (0).

Example 2: Find the zeros of:  In some problems, one or more of the factors will appear more than once when the function is
factored. The power of a factor is called its multiplicity. So given then
the multiplicity of the first factor is 3, the multiplicity of the second factor is and the multiplicity of
the third factor is 1.

Description of the Behavior at Each x-intercept

1. Even Multiplicity: The graph touches the x-axis, but does not cross it. It looks like a parabola
there.
2. Multiplicity of 1: The graph crosses the x-axis. It looks like a line there.
3. Odd Multiplicity greater than or equal 3: The graph crosses the x-axis. It looks like a cubic there.

You can use all of this information to generate the graph of a polynomial function.
• degree of the function
• end behavior of the function
• x and y intercepts (and multiplicities)
• behavior of the function through each of the x intercepts (zeros) of the function

Steps to graphing other polynomials:
1. Determine the leading term. Is the degree even or odd? Is the sign of the leading coefficient
positive or negative?
2. Determine the end behavior. Which one of the 4 cases will it look like on the ends?
3. Factor the polynomial.
4. Make a table listing the factors, x intercepts, multiplicity, and describe the behavior at each x
intercept.
5. Find the y- intercept.
6. 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 x and y intercepts. State the degree of the function. Sketch the graph of  Example 4: Find the x and y intercepts. State the degree of the function. Sketch the graph of  Example 5: Find the x and y intercepts. State the degree of the function. Sketch the graph of  Example 6: Find the x and y intercepts. State the degree of the function. Sketch the graph of  Example 7: Write the equation of the cubic polynomial P(x) that satisfies the following conditions:
zeros at x = 3, x = -1, and x = 4 and passes through the point (-3, 7)

Example 8: Write the equation of the quartic function with y intercept 4 which is tangent to the x
axis at the points (-1, 0) and (1, 0).

Math 1310
Section 4.2: Dividing Polynomials

In this section, you’ll learn two methods for dividing polynomials, long division and synthetic
division. You’ll also learn two theorems that will allow you to interpret results when you divide.

Suppose P(x) and D(x) are polynomial functions and D(x) ¹ 0. Then there are unique polynomials
Q(x) (called the quotient) and R(x) (called the remainder) such that P(x) = D(x) *Q(x) + R(x) .

We call D(x) the divisor. The remainder function, R(x) , is either 0 or of degree less than the degree
of the divisor.

You can find the quotient and remainder using long division. Recall the steps you
learned in elementary school to perform long division:

Example 1:
Divide Example 2: Divide Example 3: If and , in .

Often it will be more convenient to use synthetic division to divide polynomials. This method is
easy to use, as long as your divisor is x ± c , for any real number c.

Dividing Polynomials Using Synthetic Division

Example 4:
Divide using synthetic division Example 5: Divide using synthetic division Example 6: Divide using synthetic division Here are two theorems that can be helpful when working with polynomials:

The Remainder Theorem: If P(x) is divided by x - c, then the remainder is P(c) .

The Factor Theorem: c is a zero of P(x) if and only if x - c is a factor of P(x), that is if the
remainder when dividing by x - c is zero.

You can use synthetic division and the remainder theorem to evaluate a function at a given value.

Example 7: Use synthetic division and the remainder theorem to find P(3) for Example 8: Determine if x + 2 is a factor of And you may also need to work backwards.

Example 9: Find a polynomial with a degree of 4 with zeros at -3, 0, 2, 5.

Example 10: Find a polynomial of degree 3 with zeros at 0, 2 and -3.

Math 1310
Section 4.3: Roots of Polynomial Functions

You’ll need to be able to find all of the zeros of a polynomial. You’ll now be expected to find both
real and complex zeros of a function.

A polynomial of degree n ³ 1has exactly n zeros, counting all multiplicities.

To find all zeros, you’ll factor completely. From the factored form of your polynomial, you’ll be
able to read off all the zeros of the function.

If c is a zero of a polynomial P, then x = c is a root of the equation P(x) = 0.

If your polynomial has real coefficients, then the polynomial may have complex roots. Complex
roots occur in pairs, called complex conjugate pairs. This means that if a + bi is a root of P then so
is a - b.

Note: Example 1: Find the zeros of the polynomial write the polynomial in factored form and then state
the multiplicity of each zero. (Sometimes it may be easier to factor the polynomial first, then find
the zeros.)    You can also work backwards to writing a polynomial with integer coefficients that meets stated
conditions.

Example 2: Find a 3rd degree polynomial with integer coefficients given -5, and i are zeros

Example 3: Find a polynomial with integer coefficients given the zeros at 2 and 2 - 5i.

Example 4: Write a polynomial with integer coefficients with degree 4 and zeros at -3
(multiplicity 2) and - 3i .

Example 5: Write a polynomial with integer coefficients with degree 3 and zeros at 5 and 4 + i
with a constant coefficient of 170.