Ch.3 Polynomial and Rational

Functions

Rachel JohnsonBrittany Stephens

End Behavior• A description of what happens

as x becomes large in the positive or negative direction

• To describe, use the following notationx→ ∞ means “x becomes large in the positive direction”

x → -∞ means “x becomes large in the negative direction”

Graphing Polynomial Functions

1.Zeros. Factor the polynomial to find all its real zeros; these are the x-intercepts of the graph.

2.Test Points. Make a table of values for the polynomial.

Graphing Polynomial Functions

3. End Behavior. Determine the end behavior of the polynomial.

4. Graph. Plot the intercepts and other points you found in the table. Sketch a smooth curve that passes through these points and exhibits the required end behavior.

Long Division

•Long division for polynomials is very much like long division for numbers.

Example

122664 2 xxx

26 x

xx 246 2

82

122

x

x

4

Solution

44

2

264

12266

xxx

xx

Synthetic Division

•A quick method of dividing polynomials

•Can be used when the divisor is in the form x-c

•Ex. 3

572 23

x

xx

Example 3 | 2 -7 0 5

6 -3 -9

2 -1 -3 -4

Solution:3

432 2

x

xx

Rational Zero Theorem

•p=factors of constant term •q=factors of leading

coefficient•Every rational zero of P(x) is

in the form

01

1 ...)( axaxaxP nn

nn

0a

q

p

Finding the Rational Zeros of a Polynomial

1. List Possible Zeros. List all possible rational zeros using the Rational Zeros Theorem.

2. Divide. Use synthetic division to evaluate the polynomial at each of the candidates for rational zeros that found in Step 1. When the remainder is 0, note the quotient you have obtained.

Finding the Rational Zeros of a Polynomial

3. Repeat. Repeat Steps 1 and 2 for the quotient. Stop when you reach a quotient that is quadratic or factors easily, and use the quadratic formula or factor to find the remaining zeros.

Complex Zeros• Once you have found all the rational

zeros of an equation, you might have a quadratic equation left over from factoring.

• To find its zeros, use the quadratic formula

• Then, place the answer in a+bi form.

a

acbb

2

42

Sketching Graphs of Rational Functions

1. Factor. Factor the numerator and denominator.

2. Intercepts. Find the x-intercepts by determining the zeros of the numerator, and the y-intercept from the value of the function of x=0.

Sketching Graphs of Rational Functions

3. Vertical Asymptotes. Find the vertical asymptotes by determining the zeros of the denominator, and then see if y → ∞ or y → -∞ on each side of every vertical asymptote.

4. Horizontal Asymptote. Find the horizontal asymptote (if any) by dividing both numerator and denominator by the highest power of x that appears in the denominator, and then letting x → ∞

Sketching Graphs of Rational Functions

5. Sketch the Graph. Graph the information provided by the first four steps. Then plot as many additional points as needed to fill in the rest of the graph of the function.

Asymptotes

1. The vertical asymptotes of r are the lines x=a, where a is a zero of the denominator.

2. (a) If n<m, then r has horizontal asymptote y=0.(b) If n=m, then r has horizontal asymptote y=

(c) If n>m, then r has no horizontal asymptote.

011

1

011

1

...

...)(

bxbxbxb

axaxaxaxr

mm

mm

nn

xn

m

n

b

a