Find Rational Zeros, I

Objectives:

1. To find the zeros of a polynomial function

2. To use the Rational Zero Theorem to find the possible rational zeros of a polynomial function

Objective 1

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

Exercise 1

Use synthetic division to divide f(x) = 2x3 – 11x2 + 3x + 36 by x – 3.

Exercise 1

Use synthetic division to divide f(x) = 2x3 – 11x2 + 3x + 36 by x – 3.

Since the remainder is zero when dividing f(x) by x – 3, we can write:

This means that x – 3 is a factor of f(x).

2( )2 5 12,

3

f xx x

x

2 so ( ) ( 3)(2 5 12)f x x x x

Factor Theorem

A polynomial f(x) has a factor x – k if and only if f(k) = 0.

This theorem can be used to help factor/solve a polynomial function if you already know one of

the factors.

Exercise 2

Factor f(x) = 2x3 – 11x2 + 3x + 36 given that x – 3 is one factor of f(x). Then find the zeros of f(x).

Exercise 2

Factor f(x) = 2x3 – 11x2 + 3x + 36 given that x – 3 is one factor of f(x). Then find the zeros of f(x).

Factoring this polynomial was not

too laborious, since we already knew one of the

factors.

How are we supposed to factor a polynomial,

though, if we don’t know any factors with

which to start?

Exercise 3

Factor each of the following polynomial functions completely given that x – 4 is a factor. Then find the zeros of f(x).

1. 2. 3 2( ) 6 5 12f x x x x 3 2( ) 22 40f x x x x

Exercise 4

Find the other zeros of f(x) given that f(-2) = 0.

1. 2. 3 2( ) 2 9 18f x x x x 3 2( ) 8 5 14f x x x x

Objective 2

You will be able to use the Rational Zero Theorem to find the possible zeros of a polynomial function

pq

Exercise 5

Find the other zeros of f(x) = 10x3 – 81x2 + 71x + 42 given that f(7) = 0.

Exercise 5

Find the other zeros of f(x) = 10x3 – 81x2 + 71x + 42 given that f(7) = 0.

Writing each zero as a rational number, we have:

Notice that the numerators are factors of 42 and the denominators are factors of 10

7,

1

3,

2

2and

5

Factors of 42, the constant term.

Factors of 10, the leading coefficient.

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 Test. Consider the function below:

In factored form:

And here are the zeros:

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

5 2, 3, and

2 3x

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

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

x3

5

Rational Zero Theorem

To help you remember the order of the factors when using the Rational Zero Theorem, consider the linear function:

To find the zero(s), let y = 0 and solve for x.

6 10y x

0 6 10x

Factor of -10, the constant term.

Factor of 6, the leading coefficient.

x610

x3

5

Rational Zero Theorem

When solving this equation, first you add/subtract the constant term to the opposite side of the equation, and then you divide by the leading coefficient.

0 6 10x

Factor of -10, the constant term.

Factor of 6, the leading coefficient.

x610

Exercise 6

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

1. 2. 3 2( ) 3 6f x x x x 3 2( ) 2 7 4 8f x x x x

Exercise 7

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

1. 2. 3 2( ) 9 23 15f x x x x 3 2( ) 2 3 11 6f x x x x

Exercise 8

Once you have applied the Rational Zero Theorem to generate a list of possible rational zeros for a given polynomial function, how could you use said list to help you factor that polynomial?

Verifying Zeros

The Rational Zero Theorem only lets you find a possible list of zeros for a particular function. Now you have to use synthetic division to verify which in your list are actually zeros of the function.

Once you find one zero, then you can simplify your problem using the Factor Theorem

Exercise 9

Find all real zeros of the function.3 2( ) 4 15 18f x x x x

Exercise 10

Find all real zeros of each function.

1. 2. 3 2( ) 8 5 14f x x x x 3 2( ) 5 7 35f x x x x

5.6: Find Rational Zeros, I

Objectives:

1. To find the zeros of a polynomial function

2. To use the Rational Zero Theorem to find the possible rational zeros of a polynomial function