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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
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Polynomial Inequalities
x3 + 3x2 – x – 3 < 0
dashed
( , 3) ( 1,1)
Where is the
polynomial less
than 0?
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Let’s do this without having to graph it first.
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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.
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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)
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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
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You do this one……
2x3 – 3x2 – 32x > - 48
3( 4, ) (4, )
2
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Rational inequalities
Find the solution to a rational inequality algebraically.
40
2
x
x
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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
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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, )
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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!
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2 73
5
x
x
You do this one……
( ,5) [8, )
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You do this one……
( , 14] (6, )
5 3
6 2x x
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You do this one……
33
1 4
x x
x x
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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
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HW: WS 4-5