Polynomials and Rational Functions

By Lane Gonzalez, Alexus Brown, Helena Chevalier

Polynomial Degree

011

1 ...)( axaxaxaxP nn

nn

• Where n is a nonnegative integer and • The numbers are called the coefficients of

the polynomial• is the constant term• is the leading coefficient.

0nanaaaa ,...,,, 210

0a

na

Finding rational zeros

2414)( 23 xxxxP

• Finding the real zeros of a polynomial can be done by:– factoring the polynomial algebraically– Using the Rational Zeros Theorem/Quadratic Formula– Graphing the Function

)4)(3)(2()( xxxxP

4,3,2 x

Algebraically

65)( 2 xxxf

)2)(3()( xxxf

2,3x

Factor!

Rational Zeros!

THAT WAS FUN!!

Finding Real Zeros

• List possible zeros

• Divide

• Repeat

List all possible zerosQ

• P/Q= Factors of P divided by Factors of Q

6132)( 23 xxxxfP

2

6,2

3,2

2,2

1,1

6,1

3,1

2,1

1 Simplify

2

3,2

1,6,3,2,1

Substituting and Dividing

• Substitute possible zeros one at a time.

46)1(13)1()1(2)1( 23 f 1 is not a zero of “f”

06)2(13)2()2(2)2( 23 f 2 is a zero of “f”

Now synthetically divide by 2 (see board)

646 x

)42)(2)(42)(2( 22 xxxxxx

)8)(8( 33 xx

Using the quadratic formula for each part of the now factored equation, the zeros are:

31,31,31,31,2,2 iiii

Finding Complex Zeros

End Behavior

• End behavior is determined by the degree of the leading coefficient

• If it has odd degree

Dividing Polynomials

• Dividing polynomials is a lot like dividing numbers

122664 2 xxx

See Board

Graphing Polynomial Functions

• To graph a polynomial function– Find the zeros– Get some test points– Determine end behavior– Graph

Example

)32()( 2 xxxxf

xxxxf 32)( 23

)1)(3()( xxxxf

X=0,3,-1

Example cont.

x f(x)

-3 -20

-2 0

-1 6

0 4

1 0

3/2 -7/8

2 0

3 10

Graph on calculator now!

Graphing Rational Functions

• Most easily done by putting directly into calculator

Example:

f(x)=1/x