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5.1 Polynomial Functions

In this section, we will study the following topics:

Identifying polynomial functions and their degree

Determining end behavior of polynomial graphs

Finding real zeros of polynomial functionsand their multiplicity

Analyzing the graph of a polynomial function

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Definition of Polynomial Function

In polynomial functions, THE EXPONENTS ON THE VARIABLE CANNOT BE FRACTIONS AND CANNOT BE NEGATIVE.

Definition of Polynomial Function

Let n be a nonnegative integer and let an, an-1, …, a2, a1, a0 be real numbers.

The following function is called a polynomial function of x with degree n.

1 21 2 1 0( ) ...n n

n nf x a x a x a x a x a

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Graphs of Polynomial Functions

4 out of 5 math students surveyed preferred polynomial functions over the next leading brand of function (because their properties make them easy to analyze).

One of the most important features of polynomial functions is the fact that their GRAPHS are SMOOTH and CONTINUOUS. (This fact is very important in Calculus.)

For now, this fact helps us to find the zeros of polynomial functions and to sketch their graphs fairly well by hand.

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Polynomials are often classified by their degree. The degree of a polynomial is the highest degree of its terms.

Degree Name of Polynomial Function

Example

Zero f(x) = -3

First f(x) = 2x + 5

Second f(x) = 3x2 –5x + 2

Third f(x) = x3 – 2x –1

Fourth f(x) = x4 –3x3 +7x-6

Fifth f(x) = 2x5 + 3x4 – x3 +x2

Classifying Polynomials

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Graphs of Polynomial Functions

You have already looked at the graphs of two polynomial functions in some detail: linear and quadratic.

It is VERY important to have a general idea as to the shape of the basic polynomial functions.

We can then apply transformations to the graphs to obtain new functions.

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The Leading Coefficient Test

Just as we were able to tell if a parabola would open up or down by looking at the sign of “a ” of the quadratic function, we can use the LEADING COEFFICIENT of higher degree polynomials to determine if the left and right ends of the graph rise or fall.

We call this the END BEHAVIOR of the graph of the function.

, ( ) ?

, ( ) ?

As x f x

As x f x

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Using the Leading Coefficient to Describe End Behavior: Degree is EVEN

If the degree of the polynomial is even and the leading coefficient is positive, both ends ______________.

If the degree of the polynomial is even and the leading coefficient is negative, both ends ________________.

4( )f x x 4( )f x x

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Using the Leading Coefficient to Describe End Behavior: Degree is ODD

If the degree of the polynomial is odd and the leading coefficient is positive, the graph falls to the __________ and rises to the ______________.

If the degree of the polynomial is odd and the leading coefficient is negative, the graph rises to the _________ and falls to the _______________.

5( )f x x5( )f x x

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The Leading Coefficient Test

Example

Use the leading coefficient test to describe the end behavior of the graphs of the following functions:

2( ) ( 3)f x x 3 4 7( ) 4 3 2g x x x x x

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Zeros of Polynomial Function

A nifty observation about polynomials:

1. A polynomial of degree n has at most n real zeros

2. A polynomial of degree n has at most n – 1 turning points.

For instance, a quartic polynomial (4th degree) can have AT MOST 4 real zeros (it may have less than 4 real zeros, as we will see in a later section). Also, it can have AT MOST 3 turning points. Take a look:

This quartic function has 4 real zeros and 3 turning points.

ow!

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Zeros of Polynomial Function

VERY IMPORTANT INFORMATION ABOUT REAL ZEROS!

Real Zeros of Polynomial Functions

If f is a polynomial function and r is a real number then

the following statements are equivalent.

1. x = r is a zero or root of the function.2. x = r is a solution of the equation f(x) = 0.3. (x - r) is a factor of the polynomial f(x)4. (r, 0) is an x-intercept of the graph of f.

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Zeros of Polynomial Function

r1r2 r3 r4

f

x

From the graph, we can conclude that:

1. r1, r2, r3, r4 are _____________________ of f

2. x=r1, x=r2, x=r3, x=r4 are ________________ of f(x) = 0.

3. (x-r1), (x-r2), (x-r3), (x-r4) are _____________ of f(x).

4. (r1,0), (r2,0), (r3,0), (r4,0) are _______________________________ of f

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Zeros of Polynomial Function

Example

Find all real zeros of algebraically.

Use your graphing calculator to verify your answers.

3 2( ) 20f x x x x

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Repeated Zeros

If a polynomial function has a factor in the form (x – r)m where m is a number greater than one, then we say that x = r is a repeated zero of multiplicity m.

If m is EVEN, the graph touches the x-axis at x = r, but does not cross through. (The graph turns back around at that point…the sign of f(x) does not change.)

This function has a

double zero at x = 4.

2( ) 4f x x

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Repeated Zeros

If m is ODD, the graph crosses through the x-axis at x = r (it flattens out a little bit around x = r...sign of f(x) changes.)

This function has a triple zero at x = 4.

3( ) 4f x x

Is the degree of this polynomial even or odd? ________________

Is the leading coefficient positive or negative? _________________

What is the minimum degree of this polynomial? _______________

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3 42 2 1 3f x x x x

For the polynomial, list all zeros and their multiplicities.

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Using the Zeros to Write a Polynomial Function

We can work backwards to find a polynomial function with given zeros.

For each zero, write in the form of a factor (x – r)

E.g. If the given zeros are –3, 0, and 5, write the function as

f(x) =

Multiply out the factors and write the polynomial in simplest form (in descending order)

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For example, there are an infinite number of polynomials of degree 3 whose zeros are -4, -2, and 3. They can be expressed in the form: 4 2 3f x a x x x

4 2 3f x x x x

4 2 3f x x x x

2 4 2 3f x x x x

Many correct answers

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Using the Zeros to Write a Polynomial Function

Example

Find a cubic polynomial function with the given zeros: (Verify on calculator)

22, , 3

3

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Using the Zeros to Write a Polynomial Function

Example

Find a cubic polynomial function with the given zeros: -5; 1 multiplicity 2 (Verify on calculator)

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Sketching Polynomial Functions by Hand

Follow these steps when sketching polynomial functions:

1. Determine the BASIC SHAPE of the graph. You can tell by the degree of the polynomial.

2. Use the leading coefficient to determine the END BEHAVIOR of the

graph (whether graph falls or rises as x approaches and - ).

3. Find the REAL ZEROS of the polynomial, if possible. These will be the x-intercepts of the graph. Watch out for repeated zeros.

4. Plot a few ADDITIONAL POINTS by substituting a few different values of x into the original function. Include some points close to the x-intercepts.

5. SKETCH the graph, keeping in mind that graphs of polynomials are smooth and continuous.

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(Continued)

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Sketching Polynomial Functions by Hand

Example

Sketch the graph of by hand.4 3 2( ) 2 2 12f x x x x

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Sketching Polynomial Functions by Hand4 3 2( ) 2 2 12f x x x x

y-axis: count by 4

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Using the graph to write a polynomial function

Example

Write the equation for the cubic function shown in the graph below.

(0, 5)

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Using the graph to write a polynomial function

Example

Write the equation for the quartic function shown in the graph below.

(0, -3)

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End of Section 5.1