5.6 Notes: Find Rational Zeros

Rational Zeros: Where the graph crosses the x-axis at a rational number

Rational Zero Theorem:

To find the possible rational zeros take the factors of the last number and divide it by the factors of the first number, using positive and negative.

This helps narrow down the possible zeros, allowing us to divide to find the other factors

Steps to find the rational zeros:

1. List the possible zeros

2. Try one of the possible zeros listed

3. Keep trying until you find one that has no remainder

4. Keep factoring until you find all factors

5. Solve for all x values to find the zeros

List the possible rational zeros of f using the rational zeros theorem.

1.3 2( ) 9 23 15f x x x x

Factors of last term: 1, 15 3, 5,

Factors of first term: 1

Possible rational zeros: 1, 15 3, 5,

List the possible rational zeros of f using the rational zeros theorem.

2.3 2( ) 2 3 11 6f x x x x

Factors of last term: 1, 6 2, 3,

Factors of first term:

Possible rational zeros:

1, 2

1, 6, 2, 3, 1

,2

3

2

Find all real zeros of the function.

3. 3 2( ) 4 15 18f x x x x

Factors of last term: 1, 18 2, 9,

Factors of first term: 1

Possible rational zeros:

3, 6,

1, 18 2, 9, 3, 6,

1 1 –4 –15 18

1

1

–3

–3

–18

–18

0

(x – 1)xx

–63

(x – 1)(x – 6)(x + 3) = 0x2 – 3x – 18( ) = 0

Find all real zeros of the function.

3. 3 2( ) 4 15 18f x x x x Possible rational zeros: 1, 18 2, 9, 3, 6,

x = 1, 6, –3

Find all real zeros of the function.

4. 3 2( ) 3 19 4 12f x x x x

Factors of last term: 1, 12 2, 6,

Factors of first term: 1,

Possible rational zeros:

3, 4,

3

1, 4, 2, 3, 1

,3

2

,3

12 6, 4

3

Find all real zeros of the function.

4. 3 2( ) 3 19 4 12f x x x x Possible rational zeros: 1, 4, 2, 3,

1,

3

2,

3

12 6, 4

3

1 3 19 4 –12

3

3

22

22

26

26

14

Find all real zeros of the function.

4. 3 2( ) 3 19 4 12f x x x x Possible rational zeros: 1, 4, 2, 3,

1,

3

2,

3

12 6, 4

3

–1 3 19 4 –12

3

–3

16

–16

–12

12

0

(x + 1)3xx

–26

(x + 1)(3x – 2)(x + 6) = 03x2 + 16x – 12( ) = 0

x = 2

1, , 63

Find all real zeros of the function.

5. 3 2( ) 4 8 9 18f x x x x

Factors of last term: 1, 18 2, 9,

Factors of first term: 1,

Possible rational zeros:

3, 6,

2,

1,

2

3,

2

9

2

4

1, 18 2, 9, 3, 6,

1,

4

3,

4

9

4

Find all real zeros of the function.

5. 3 2( ) 4 8 9 18f x x x x Possible rational zeros:

1,

2

3,

2 9

2

1, 18 2, 9, 3, 6, 1

,4

3

,4

9

4

( )( )

f(x) = 4x2(x + 2) –9 (x + 2)

f(x) = (4x2 – 9)(x + 2)2x2x

3–3

(2x + 3)(2x – 3)(x + 2) = 0

x = 3 3

, , 22 2