Unit 4 – Polynomial/Rational Functions

Zeros of Polynomial Functions

(Unit 4.3)

William (Bill) Finch

Mathematics DepartmentDenton High School

Introduction Theorems Zeros Complex Zeros Summary

Lesson Goals

When you have completed this lesson you will:

I Find complex solutions to quadratic functions.

I Apply the Fundamental Theorem of Algebra.

I Find all complex zeros (real and imaginary) of polynomialfunctions.

W. Finch DHS Math Dept

Zeros 2 / 14

Introduction Theorems Zeros Complex Zeros Summary

Lesson Goals

When you have completed this lesson you will:

I Find complex solutions to quadratic functions.

I Apply the Fundamental Theorem of Algebra.

I Find all complex zeros (real and imaginary) of polynomialfunctions.

W. Finch DHS Math Dept

Zeros 2 / 14

Introduction Theorems Zeros Complex Zeros Summary

Lesson Goals

When you have completed this lesson you will:

I Find complex solutions to quadratic functions.

I Apply the Fundamental Theorem of Algebra.

I Find all complex zeros (real and imaginary) of polynomialfunctions.

W. Finch DHS Math Dept

Zeros 2 / 14

Introduction Theorems Zeros Complex Zeros Summary

Lesson Goals

When you have completed this lesson you will:

I Find complex solutions to quadratic functions.

I Apply the Fundamental Theorem of Algebra.

I Find all complex zeros (real and imaginary) of polynomialfunctions.

W. Finch DHS Math Dept

Zeros 2 / 14

Introduction Theorems Zeros Complex Zeros Summary

Fundamental Theorem of Algebra

The Fundamental Theorem of AlgebraIf f (x) is a polynomial of degree n, where n > 0, then f has atleast one zero in the complex number system.

Linear Factorization TheoremIf f (x) is a polynomial of degree n, where n > 0, then f hasprecisely n linear factors

f (x) = an(x − c1)(x − c2) · · · (x − cn)

where c1, c2, . . . , cn are complex numbers.

W. Finch DHS Math Dept

Zeros 3 / 14

Introduction Theorems Zeros Complex Zeros Summary

Fundamental Theorem of Algebra

The Fundamental Theorem of AlgebraIf f (x) is a polynomial of degree n, where n > 0, then f has atleast one zero in the complex number system.

Linear Factorization TheoremIf f (x) is a polynomial of degree n, where n > 0, then f hasprecisely n linear factors

f (x) = an(x − c1)(x − c2) · · · (x − cn)

where c1, c2, . . . , cn are complex numbers.

W. Finch DHS Math Dept

Zeros 3 / 14

Introduction Theorems Zeros Complex Zeros Summary

Zeros of a Polynomial Function

Function Degree Linear Factors Zeros

f (x) = x − 4 1 x − 4 4

g(x) = x2 − 4x + 4 2 (x − 2)(x − 2) 2, 2

h(x) = x3 + 9x 3 x(x ± 3i) 0, ±3i

j(x) = x4 − 16 4 (x ± 2)(x ± 2i) ±2, ±2i

Degree = # of Linear Factors = # of Zeros

W. Finch DHS Math Dept

Zeros 4 / 14

Introduction Theorems Zeros Complex Zeros Summary

Zeros of a Polynomial Function

Function Degree Linear Factors Zeros

f (x) = x − 4 1 x − 4 4

g(x) = x2 − 4x + 4 2 (x − 2)(x − 2) 2, 2

h(x) = x3 + 9x 3 x(x ± 3i) 0, ±3i

j(x) = x4 − 16 4 (x ± 2)(x ± 2i) ±2, ±2i

Degree = # of Linear Factors = # of Zeros

W. Finch DHS Math Dept

Zeros 4 / 14

Introduction Theorems Zeros Complex Zeros Summary

Finding the Zeros of a Polynomial Function

To find the zeros of a polynomial function you may use one ofthe following tools (or a combination of them):

I Graphing

I Factoring

I Division (long or synthetic)

In order to get started it is often helpful to identify arational zero by either graphing or with theRational Zero Test.

W. Finch DHS Math Dept

Zeros 5 / 14

Introduction Theorems Zeros Complex Zeros Summary

Finding the Zeros of a Polynomial Function

To find the zeros of a polynomial function you may use one ofthe following tools (or a combination of them):

I Graphing

I Factoring

I Division (long or synthetic)

In order to get started it is often helpful to identify arational zero by either graphing or with theRational Zero Test.

W. Finch DHS Math Dept

Zeros 5 / 14

Introduction Theorems Zeros Complex Zeros Summary

Identifying Rational Zeros

The Rational Zero TheoremIf the polynomial f (x) = anx

n + an−1xn−1 + · · ·+ a1x + a0 has

integer coefficients, every rational zero of f has the form

Rational zero =p

q

where p and q have no common factors other than ±1, and

p = an integer factor of the constant term a0

q = an integer factor of the leading coefficient an

W. Finch DHS Math Dept

Zeros 6 / 14

Introduction Theorems Zeros Complex Zeros Summary

Example 1

For the polynomial f (x) = x3 − 15x2 + 75x − 125

a) List all possible rational zeros.

b) Determine which are actually zeros of f .

W. Finch DHS Math Dept

Zeros 7 / 14

Introduction Theorems Zeros Complex Zeros Summary

Example 2

For the polynomial g(x) = 2x4 − 9x3 − 18x2 + 71x − 30

a) List all possible rational zeros.

b) Determine which are actually zeros of g .

W. Finch DHS Math Dept

Zeros 8 / 14

Introduction Theorems Zeros Complex Zeros Summary

Complex Zeros

Complex Zeros Occur in Conjugate PairsLet f (x) be a polynomial function that has real coefficients. Ifa + bi (where b 6= 0) is a zero of the function, the conjugatea − bi is also a zero of the function.

W. Finch DHS Math Dept

Zeros 9 / 14

Introduction Theorems Zeros Complex Zeros Summary

Example 3

Find all of the zeros of f (x) = x3 − 4x2 + 21x − 34 if youknow one of the zeros is 1 + 4i .

W. Finch DHS Math Dept

Zeros 10 / 14

Introduction Theorems Zeros Complex Zeros Summary

Example 4

For the polynomial h(x) = 8x3 − 4x2 + 6x − 3

a) Find all of the zeros of the function.

b) Find all of the linear factors of the function.

W. Finch DHS Math Dept

Zeros 11 / 14

Introduction Theorems Zeros Complex Zeros Summary

Example 5

Write a third-degree polynomial function whose zeros include2 and 7i .

W. Finch DHS Math Dept

Zeros 12 / 14

Introduction Theorems Zeros Complex Zeros Summary

Example 6

The water level in a bucket sitting on a patio can be modeledby

f (x) = x3 + 4x2 − 2x + 7

where f (x) is the height of the water in millimeters and x isthe time in days. On what day(s) will the water reach a heightof 10 millimeters?

W. Finch DHS Math Dept

Zeros 13 / 14

Introduction Theorems Zeros Complex Zeros Summary

What You Learned

You can now:

I Find complex solutions to quadratic functions.

I Apply the Fundamental Theorem of Algebra.

I Find all complex zeros (real and imaginary) of polynomialfunctions.

I Do problems Chap 2.4 #1-7 odd, 11, 15, 17, 33, 35, 39,49-53 odd, 61-65 odd

W. Finch DHS Math Dept

Zeros 14 / 14

Introduction Theorems Zeros Complex Zeros Summary

What You Learned

You can now:

I Find complex solutions to quadratic functions.

I Apply the Fundamental Theorem of Algebra.

I Find all complex zeros (real and imaginary) of polynomialfunctions.

I Do problems Chap 2.4 #1-7 odd, 11, 15, 17, 33, 35, 39,49-53 odd, 61-65 odd

W. Finch DHS Math Dept

Zeros 14 / 14

Introduction Theorems Zeros Complex Zeros Summary

What You Learned

You can now:

I Find complex solutions to quadratic functions.

I Apply the Fundamental Theorem of Algebra.

I Find all complex zeros (real and imaginary) of polynomialfunctions.

I Do problems Chap 2.4 #1-7 odd, 11, 15, 17, 33, 35, 39,49-53 odd, 61-65 odd

W. Finch DHS Math Dept

Zeros 14 / 14

Introduction Theorems Zeros Complex Zeros Summary

What You Learned

You can now:

I Find complex solutions to quadratic functions.

I Apply the Fundamental Theorem of Algebra.

I Find all complex zeros (real and imaginary) of polynomialfunctions.

I Do problems Chap 2.4 #1-7 odd, 11, 15, 17, 33, 35, 39,49-53 odd, 61-65 odd

W. Finch DHS Math Dept

Zeros 14 / 14

Introduction Theorems Zeros Complex Zeros Summary

What You Learned

You can now:

I Find complex solutions to quadratic functions.

I Apply the Fundamental Theorem of Algebra.

I Find all complex zeros (real and imaginary) of polynomialfunctions.

I Do problems Chap 2.4 #1-7 odd, 11, 15, 17, 33, 35, 39,49-53 odd, 61-65 odd

W. Finch DHS Math Dept

Zeros 14 / 14