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5.5 Real Zeros of Polynomial Functions

In this section, we will study the following topics:

The Remainder and Factor Theorems

The Rational Zeros Theorem

Finding the Real Zeros of a Polynomial Function

Solving Polynomial Equations

The Intermediate Value Theorem

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Review Topics

For the next two sections, you will need to know the following:

1. Polynomial Long Division (pp 44-47)

2. Synthetic Division (pp 57-60)

3. Quadratic Formula (pp 102-104)

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Example

Review: Long Division of Polynomials

3Divide 1 by 1x x x

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Remember, for every division problem the following statement is true:

Dividend = (divisor) x (quotient) + remainder

Long Division of Polynomials

2. . 17 5 3 17 5 3 2re g

For polynomial division, we express the division algorithm as:

( ) ( ) ( ) ( )f x d x q x r x

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Long Division of Polynomials

( ) ( ) ( ) ( )f x d x q x r x

Using the previous example, we have

?

3 21 1 1x x x x x

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Synthetic division is a really cool shortcut for dividing polynomials by divisors in the form x – c.

As with long division, before you start synthetic division, first do the following:

1. Arrange the terms in descending order. 2. Use zero placeholders, where necessary.

NOTE: If you are dividing by the binomial x - c, then you would use c as the divisor.

Likewise, if you are dividing by x + c, then you would use (- c) as the divisor, since x + c = x - (- c).

Review: Synthetic Division

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Synthetic Division

3 25 4 5 3x x x x Example #1

Divide

The divisor is x – 3, so c =

Arranging the terms of the dividend in descending order we have:

So, you would set this synthetic division problem up as:

c Coefficients of dividend

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Synthetic Division

Example #1 (cont)

Now we are ready to divide.

3 1 4 5 5

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Synthetic Division (continued)

Example #2

Use synthetic division to divide: 35 6 8 2x x x

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Quadratic Formula

To solve equations in the form:

2 4

2

b b acx

a

2 0ax bx c

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The Remainder Theorem

This means that you can evaluate a polynomial function at a given value by -substituting the value into the function,

OR

-using synthetic division to find the remainder.

The Remainder Theorem

If a polynomial f(x) is divided by x – c, the remainder r is the same as the value of f(c).

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The Remainder Theorem

Example

Find f(5) given f(x) = 4x2 –10x – 21

Old, boring method: Exciting new synthetic synthetic method:

f(5) = 4(5)2 –10(5) – 21

f(5) =

f(5) =

f(5) =

5 4 10 21

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The Factor Theorem

The Factor Theorem

If a polynomial f(c) has a factor (x – c) iff f(c) = 0.

This theorem tells us that, in order for x – c to be a factor of the polynomial, the remainder when the polynomial is divided by x – c must be zero.

That makes sense...

For example, 8 is a factor of 32 since 8 divides into 32 evenly (the remainder is zero.)

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The Importance of the Remainder Theorem

If f(x) is divided by x – c and the remainder is equal to 0, then

(x – c) is a factor of f(x)

If c is a real number, (c, 0) is an x-intercept of the graph of f.

So, from this graph of f(x), we can determine:

3 2

Use the Factor Theorem to determine whether the function-2 - 4 3 has the factor

(a) 1 (b) 1

f x x x x

x x

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The Rational Zero Theorem

The Rational Zero Theorem

If the polynomial f(x)=anxn + an-1xn-1 + … + a2x2 + a1x + a0 has

integer coefficients, then every rational zero must have the form ,

where

p IS A FACTOR OF THE CONSTANT TERM a0 , and

q IS A FACTOR OF THE LEADING COEFFICIENT an.

This theorem will enable us to list all of the potential rational zeros of a polynomial, using the form

factors of the constant ter

factorsPotential ra

of the leadtiona

ing cl zeros =

oeffi

m

cient

p

q

3 2

List the potential rational zeros of3 8 7 12f x x x x

Factors of the constant

Factors of the leading coefficient

: 1, 2, 3, 4, 6, 12p : 1, 3q

1 2 4: 1, 2, 3, 4, 6, 12, , ,

3 3 3

p

q

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The Rational Zero Test

factors of the constant term

factors oPossible

f the learati

dingonal

coefzeros =

ficient

Example

Find all potential rational zeros of 4 3 2( ) 2 17 35 9 45f x x x x x

p

q

Solution

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The Rational Zero Test (continued)

Once you have the list of all potential rational zeros, you need

to use trial and error to test them using synthetic division (or by

substituting them into the function) to determine which ones are

actual zeros. Remember, the remainder (or the functional

value) must be equal to zero.

A Sneaky Technology Shortcut:

You can use the graph or the table of values to find one or more

of the rational zeros, if there are any. Use the fact that a real

zero is an x-intercept of the graph.

Use this zero to perform synthetic division. A zero remainder

will confirm that it is an actual zero.

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The Rational Zero Test (continued)

Example

Use the Rational Zero Test to find ALL rational zeros of 4 2( ) 4 17 4f x x x

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Using the Quotient to find the remaining zeros

Each time you divide a polynomial using synthetic division, the

QUOTIENT HAS A DEGREE THAT IS ONE LESS THAN THE

ORIGINAL POLYNOMIAL.

We will use the resulting lower-degree (“depressed”) polynomial

find the remaining zeros.

Your goal is obtain a lower-degree polynomial that is

quadratic. Then you can find the remaining zeros by factoring

or using the quadratic formula.

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FIND ONE RATIONAL ZERO, using the Rational Zero Theorem to

find potential rational zeros and then using the graph to help you

locate one rational zero.

DIVIDE THE CUBIC POLYNOMIAL BY THE RATIONAL ZERO

using synthetic division.

SOLVE THE DEPRESSED QUADRATIC EQUATION to find the

remaining zeros (by factoring, completing the square, or quadratic

formula).

Scenario 1: The original polynomial is CUBIC

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Find all real zeros of algebraically.3 2( ) 3 16 12f x x x x

Example #1

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Find all real zeros of algebraically.3 2( ) 2 2 5 5f x x x x

Example #2

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FIND TWO RATIONAL ZEROS, using the rational zero test to find

potential rational zeros and then using the graph to help you locate

TWO rational zeros. (Be on the alert for double zeros!)

DIVIDE THE QUARTIC POLYNOMIAL BY ONE OF THE RATIONAL

ZEROS using synthetic division.

DIVIDE THE DEPRESSED CUBIC POLYNOMIAL BY THE OTHER

RATIONAL ZERO using synthetic division.

SOLVE THE DEPRESSED QUADRATIC EQUATION to find the

remaining zeros (by factoring, completing the square, or quadratic

formula).

Scenario 2: The original polynomial is QUARTIC

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Find all real zeros of algebraically.4 3 2( ) 8 14 16 32f x x x x x

4 3 2( ) 8 14 16 32f x x x x x

Example #1

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Show that -5 and 3 are zeros of f(x) and use this information to write

the complete factorization of f.

4 3 2( ) 2 7 26 49 30f x x x x x

Example #2

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Approximating Zeros of Polynomial Functions

Often we are not able to find the zeros of a polynomial function algebraically, but we can still approximate the value of the zero.

We know that the graphs of polynomials are continuous; therefore, if the sign of the function values (y-values) changes from negative to positive or vice-versa, we know that the graph must have passed through the x-axis and hence, has a zero in that interval.

This result stems from the INTERMEDIATE VALUE THEOREM.

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Approximating Zeros of Polynomial Functions

We can use the table of values to find the intervals (of length 1) in which a polynomial function is guaranteed to have a zero.

The functional (y) values change:

from + to – in the interval -4 < x < -3,

from – to + in the interval –1 < x < 0,

from + to – in the interval 0 < x < 1.

So this polynomial function is guaranteed to have a zero in each of the following intervals: (-4, -3), (-1, 0), and (0, 1).

Example:

The following is a table of polynomial function values.

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Approximating Zeros of Polynomial Functions

Example:

Use the table of values on your calculator to find the intervals (of length 1) in which the function is guaranteed to have a zero.

3 2( ) 5 3f x x x

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End of Sect. 5.5