Lesson 4-1

Polynomial Functions

Vocabulary

• Polynomial in one variable – an expression of the form . The coefficients represent complex numbers (real or imaginary) is not zero, and n represents a nonnegative integer.

• Degree – the greatest exponent of a polynomials variable.

• Leading coefficient – the coefficient of the variable with the greatest exponent.

• Polynomial function – a function where P(x) is a polynomial in one variable.

• Polynomial equation - a polynomial that is set equal to zero.• Root – a solution of the equation P(x) = 0• Imaginary Number – a complex number of the form where

and is the imaginary number.• Complex numbers – the imaginary numbers combined with

the real numbers• Pure Imaginary Number – the complex number when a = 0

and • Fundamental Theorem of Algebra – Every polynomial

equation with degree greater than zero has at least one root in the set of complex numbers

Polynomial Functions

State the degree and the leading coefficient of the polynomial

13)( 234 xxxxf

Degree of 4 and leading coefficient of 3Determine whether -2 is a zero of f(x) f(-2) = 59 Not a zero

State the degree and the leading coefficient of the polynomial

1835)( 23 xxxxf

Degree of 3 and leading coefficient of 1Determine whether 6 is a zero of f(x) f(6) = 0 Yes it is a zero

State the degree and the leading coefficient of the polynomial

285)( 52 xxxf

Degree of 5 and leading coefficient of 8Determine whether 1 is a zero of f(x) f(1) = 15 No

Example 2

Fundamental Theorem of Algebra

Corollary to the Fundamental Theorem of Algebra

Example 3

Write a polynomial equation of least degree with roots 2, 3i, and -3i

01892

0)9)(2(

0)9)(2(

0)3)(3)(2(

23

2

22

xxx

xx

ixx

ixixx

Write a polynomial equation of least degree with roots -5 and 7

0352

0)7)(5(2

xx

xx

Write a polynomial equation of least degree with roots 6, 2i, -2i, i,-i

02443056

0)45)(6(

0)1)(4)(6(

0))(4)(6(

0))()(2)(2)(6(

2345

24

22

2222

xxxxx

xxx

xxx

ixixx

ixixixixx

Example 3

Does the equation have an odd or even degree? How many times does the graph of

the related function cross the x-axis.

01892 23 xxx

Odd, once

Does the equation have an odd or even degree? How many times does the graph of

the related function cross the x-axis.

03522 xxEven, twice

Does the equation have an odd or even degree? How many times does the graph of

the related function cross the x-axis.

02443056 2345 xxxxx

Odd, once

State the number of complex roots of the equation has. Then find the roots and

graph the related function.

04359 24 xx

The polynomial has a degree of 4 so there are 4 complex roots.

2 real roots and 2 imaginary roots for a total of 4 complex roots

Graph with the graphing Calculator

04359 24 xx

State the number of complex roots of the equation has. Then find the roots and

graph the related function.

0443232 23 xxx

3 Complex Roots

Graph on graphing calculator

Total of 3 Complex Roots,2 imaginary, 1 real

4

2

32

4

432

)1)(432(

0)1(4)1(32

2

2

2

2

ix

x

x

xx

xxx

x-1=0x=1

0443232 23 xxx

State the number of complex roots of the equation has. Then find the roots and

graph the related function.

049142 xx

2 Complex Roots

Graph on graphing calculator

Total of 2 Complex Roots,2 real (double root)

x-7=0x=77

0)7(

)7)(7(

049142

x

x

xx

xx049142 xx

State the number of complex roots of the equation has. Then find the roots and

graph the related function.

082 23 xxx

3 Complex Roots

Graph on graphing calculator

3 complex roots3 real roots

0)2)(4(

0)82(

0822

23

xxx

xxx

xxx

x=0, x=-4, x=2

Example 5