Find Rational Zeros, II

Objectives:

1. To use a graph to help find the zeros of a polynomial function

2. To use the Location Principle to help find the zeros of a polynomial function

Warm-Up

The points (2, −3) and (3, 5) are on the graph of a polynomial function f (x). What conclusion can you draw about the graph of f (x)?

There must be a zero between 2 and 3 since the graph must cross the x-axis to connect these points.

The Rational Zero Theorem

If f(x) = anxn + … + a1x + a0 has integer coefficients, then every rational zero of f(x) has the following form:

Important note: These factors can be either positive or negative.

0factor of constant term

factor of leading coefficient n

a

q

p

a

The Rational Zero Theorem

Here’s another way to think about The Rational Zero Theorem. Consider the function below:

In factored form:

And here are the zeros:

( ) (2 5)( 3)(3 2)f x x x x

3 2( ) 6 47 30f x x x x

5 2, 3, and

2 3x

The Rational Zero Theorem

Now work backwards from the factors: What would be the constant term and the leading coefficient?

( ) (2 5)( 3)(3 2)f x x x x

3 2( ) 6 47 30f x x x x

5 2, 3, and

2 3x

3(2 )( )(3 ) 6x x x x (5)( 3)(2) 30

Factors of 6 Factors of −30

The Rational Zero Theorem

Remember that The Rational Zero Theorem just helps you to generate a list of the possible rational zeros of a polynomial function (with integer coefficients).

To verify each zero, you have to use synthetic division.

Exercise 1

List the possible rational zeros of f(x) using the Rational Zero Theorem.

3 2( ) 12 35 24f x x x x

Exercise 2

Find all real zeros of the function.

3 2( ) 12 35 24f x x x x

Exercise 3

List the possible rational zeros of f(x) using the Rational Zero Theorem.

3 2( ) 2 5 11 14f x x x x

Exercise 3

List the possible rational zeros of f(x) using the Rational Zero Theorem.

Notice that when the leading coefficient is not 1, the number of possible rational zeros increases in an unsavory way. Surely there has to be a way to narrow the possibilities…

3 2( ) 2 5 11 14f x x x x

Exercise 4

According to The Rational Zero Theorem, what is the maximum number of possible rational zeros for the polynomial function below?

3 2( )f x x bx cx d

Exercise 5

Assume a has k factors and d has r factors. According to The Rational Zero Theorem, what is the maximum number of possible rational zeros for the polynomial function below?

3 2( )f x ax bx cx d

Objective 1

You will be able to use a graph to help find the zeros of a polynomial function

Facilitation, 1 (Cheating)

The first way to facilitate the search for rational zeros involves looking at the graph of the polynomial.

Facilitation, 1 (Cheating)

Since zeros are the same thing as a graph’s x-intercepts, finding the approximate location of an x-intercept can give a reasonable place to start verifying factors from our long, tedious list of possibilities.

Exercise 6

Find all real zeros of the function.3 2( ) 2 5 11 14f x x x x

Facilitation, 1 (Cheating)

Using the graph of a polynomial function to help finding its zeros seems a bit like cheating. This is because we want to find the zeros (or x-intercepts), and we do this by looking a graph’s x-intercepts. If we already have a graph, then why are we trying to find the graph’s x-intercepts algebraically?

Graph

Zeros

(A fine example of circular logic)

Exercise 7

Find all real zeros of the function.3 2( ) 48 4 20 3f x x x x

Exercise 8

Find all real zeros of the function.4 3 2( ) 2 5 18 19 42f x x x x x

Objective 2You will be able to use the Location Principle to help find the zeros of a polynomial function

Location Principle

Another way to narrow down our rational zero choices is by using the Location Principle. Consider the table of values. What can you conclude about the location of one of the zeros of f(x)?

0 −61 −122 283 1504 3905 784

There must be a zero here

Location Principle

Location Principle

If is a polynomial function and and are two numbers such that and ,

then has at least one zero between

and .

For example, if and , then there must

be a zero between and . This is because a

continuous curve would cross the -axis when

connecting these points.

Location Principle

Location Principle

There must be a zero between 2 and 3 since the graph must cross the x-axis to connect these points.

If is a polynomial function and and are two numbers such that and ,

then has at least one zero between

and .

Location Principle

Location Principle

If is a polynomial function and and are two numbers such that and ,

then has at least one zero between

and .

So we could use a table of values with

the Location Principle to narrow our rational

zero choices. Just look for where the

function-values change signs: + to –

or – to +.

Exercise 9

Find all real zeros of the function.4 3 2( ) 24 38 191 157 28f x x x x x

Exercise 10

The Rational Zero Theorem only helps in finding rational zeros. How would we go about finding the irrational zeros of a polynomial function?

Once it comes down to an unfactorable quadratic, use the quadratic formula or

completing the square to find the last two zeros.

Exercise 11

Find all real zeros of the function.4 3 2( ) 9 3 30 6 12f x x x x x

Exercise 12

Find all real zeros of the function.4 3 2( ) 10 11 42 7 12f x x x x x

Find Rational Zeros, II

Objectives:

1. To use a graph to help find the zeros of a polynomial function

2. To use the Location Principle to help find the zeros of a polynomial function