7-5 Roots and Zeros 7-6 Rational Zero Theorem Glencoe – Algebra 2 Chapter 7: Polynomial Functions 1 Skills: Determine the number and type of roots for.

7-5 Roots and Zeros7-6 Rational Zero

Theorem

Glencoe – Algebra 2Chapter 7: Polynomial Functions

1

Skills: • Determine the number and type of roots

for a polynomial equation.• Find the zeros of a polynomial function.• Identify the possible rational zeros of a

polynomial function.• Find all rational zeros of a polynomial

function.

2

Fundamental Theorem of AlgebraIf f(x) is a polynomial of degree n where n>0, then the equation f(x)=0 has at least one complex solution.

CorollaryIf f(x) is a polynomial of degree n where n>0, then the equation f(x)=0 has exactly n solutions, including doubles/triples.

Behavior Near ZerosWhen a factor x – k is raised to an odd

power (single/triple zero), the graph crosses the x-axis at x = k.

When a factor x – k is raised to an even power (double/quadruple zero), the graph touches (is tangent to) the x-axis at x = k.

Complex Conjugates TheoremIf a + bi is a zero,

then so is its conjugate a – bi.

Irrational Conjugates TheoremIf is a zero,

then so is its conjugate .a b

a b

Descartes’ Rule of SignsPositive Real Zeros: # of sign changes in

f(x) or less by an even numberNegative Real Zeros: # of sign

changes in f(-x) or less by an even number

Glencoe – Algebra 2Chapter 7: Polynomial Functions

3

Rational Zero Theorem(Rational Roots Theorem)

If the polynomial has integer coefficients, then all of the rational roots will be of the form:

Finding Rational Roots (Zeros)1.Use the rational zero theorem to find all possible roots.2.Test the possible roots by plugging them into the equation until you get one root.3.If the answer you get is zero, then the number is a root.4.Use synthetic division to break the polynomial down into factors.5.Solve using the factor theorem to find the roots.6.If you’re having trouble factoring, test other possible roots in the polynomial you’re trying to factor again and use synthetic division.

f actor of constant term

f actor of leading coeffi cient

p

q

Glencoe – Algebra 2Chapter 7: Polynomial Functions

4

Steps for Finding All Zeros 1. Use Descartes’ Rule of

Signs to make a zeros table.2. Use the Rational Zeros

Theorem to find all possible rational zeros.

3. Use synthetic division to test the zeros and break down the polynomial.

4. If you can, factor.5. Use the quadratic formula

to find irrational/imaginary zeros.

6. Remember irrational/imaginary zeros come in pairs (conjugates).

Glencoe – Algebra 2Chapter 7: Polynomial Functions

Positive Negative Imaginary Total

3 2 0 5

3 0 2 5

1 2 2 5

1 0 4 5

1 6 15 14

1 5 9 1 14

1 4 4 10 13 14

5

Example 1Find all zeros of

5 4 3 24 4 10 13 14f x x x x x x

:Possible Zeros 1, 2, 7, 14

f x

5 1

51

1 9

9 0

Factors of 14: 1, 2, 7, 14

Factors of 1: 1

1 2 3 f x

1 2

1

1

14

14

1

61 15

14

14

1 6 15

02

1

2

4

8

7

14

0

5 4 3 24 4 10 13 14f x x x x x x

21 1 2 4 7x x x x x

2 21 2 4 7x x x x

2 42

b b acx

a

24 4 4 1 7

2 1x

4 122

4 2 32i

2 3i

The zeros are

1, 2, and 2 3.i

Glencoe – Algebra 2Chapter 7: Polynomial Functions

6

Example 2Write a polynomial function f of least

degree that has rational coefficients, a leading coefficient of 1, and the

following zeros: .3 and 2 5

: 3, 2 5,2 5Zeros

3 2 5 2 5x x x

3 2 5 2 5x x x

3 2 5 2 5x x x

223 2 5x x

23 4 4 5x x x

23 4 1x x x

3x 24x x 23x 12x 3

3x 27x 11x 3

3 27 11 3f x x x x

Glencoe – Algebra 2Chapter 7: Polynomial Functions

2 11 18 9

7

Example 3Find the rational zeros of

3 22 11 18 9f x x x x

3

: 1, 3,2

Zeros

:Possible Zeros 1 3 9 1 3 9, , , , ,

1 1 1 2 2 2

Eliminate all positive numbers

because you'd get large answers.

1

1:1

3 2

1 2 1 11 1 18 1 9f

2 11 18 9

0

9 2

92

1 9

9 0

3 22 11 18 9x x x

21 2 9 9x x x

3

3 :1

2

2 3 9 3 9 0

3 2 9 9

2

6

3 0

9

1 3 2 3x x x

Factors of 9: 1, 3, 9

Factors of 2: 1, 2

Glencoe – Algebra 2Chapter 7: Polynomial Functions

10 11 42 7 12

8

Example 4Find the real zeros of

4 3 210 11 42 7 12f x x x x x

:Possible Zeros

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

1 1 1 1 1 11 2 3 4 6 12

, , , , , ,2 2 2 2 2 21 2 3 4 6 12

, , , , , ,5 5 5 5 5 51 2 3 4 6 12

, , , , ,10 10 10 10 10 10

8 5

1610

12 17

34 24

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

1 3 1 2 3, , , , ,

2 2 5 5 54 6 12 1 3

, , , ,5 5 5 10 10

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

Factors of 10: 1, 2, 5, 10

12

035

10

6

10

6

40

24

0

210 10 40 0x x

2 4 0x x

2 42

b b acx

a

1 1 4 1 4

2 1x

1 172

x

1 3 1 17

The zeros are , , and .2 5 2

Glencoe – Algebra 2Chapter 7: Polynomial Functions