November 1st copyright2009merrydavidson

Warm Up1) Complete the Square to change to

Standard Form.f(x) = 2x2 – 8x + 9

2) Find ALL roots using the rational root theorem, factoring and/or quadratic formula.

f(x) = x4 + 3x3 – 5x2 - 21x + 22

y = 2(x – 2)2 + 1

1,2, 3 2x i

Polynomial Inequalities

x3 + 3x2 – x – 3 < 0

dashed

( , 3) ( 1,1)

Where is the

polynomial less

than 0?

Let’s do this without having to graph it first.

Polynomial Inequalities

x3 + 3x2 – x – 3 < 0

Step 1: Find the CRITICAL numbers.

Solutions are critical numbers so solve for x =. x = + 1, -3

Since it is “<“ and not “<‘;

1, 3x place on the x-axis as a hole.

x3 + 3x2 – x – 3 < 0

Step 2: Test a point on each interval in the original inequality to see if it is “true” or “false”.

The solution is the intervals that turn out TRUE.

-3 -1 1

FTFT

( , 3) ( 1,1)

Process for solving polynomial inequalities……

• Pretend it is equal to zero and find zero’s

• Put those on the number line

<, > closed circles

<, > open circles

• Test sections into the ORIGINAL inequality for T or F

• Write the answer in interval notation for the true sections

You do this one……

2x3 – 3x2 – 32x > - 48

3( 4, ) (4, )

2

Rational inequalities

Find the solution to a rational inequality algebraically.

40

2

x

x

Solve rational inequalities

2x

40

2

x

x

Step 1: Rewrite fractions with factored denominator

Step 2: Find domain restrictions. Put this number on a number line as a “hole”.

Step 3. pretend it is = 0 and find the zero’s. Put this value on the number line as open or closed circles. <, > closed circles <, > open circles

Already done

(x-2)(x-2)

x + 4 = 0

x = -4

2-4

Solve rational inequalities

40

2

x

x

-4 2

Step 4: Test intervals using the original inequality.

Step 5. Write the solution in interval notation.

T F T

( , 4] (2, )

Process for solving rational inequalities……

• Find domain restrictions from the denominator and put a hole on the number line

• Pretend it is equal to zero and find zero’s• Put those on the number line as open/closed

dots• Test sections into the ORIGINAL inequality for T

or F• Write the answer in interval notation for the true

sectionsA common mistake is to forget to do the first step!

2 73

5

x

x

You do this one……

( ,5) [8, )

You do this one……

( , 14] (6, )

5 3

6 2x x

You do this one……

33

1 4

x x

x x

Quiz next time over….

• Find critical numbers for inequalities

• Solve polynomial and rational inequalities

• Graph a rational function including intercepts, asymptotes and correct end behavior

• Function composition

• Evaluating a function for a specific value

HW: WS 4-5