Ration Root Test

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Solve by Factoring
2
Solve by completing the Square
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Rational Root Test
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Remainder Theorem

• Remainder f(k)
• Example
• f(2)

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Find the remainder
Therefore, x -2 is NOT a root!
Factor Theorem
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Factor Theorem

• f(x) has a factor (x-k) iff f(k)0.

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Rational Zero Test

• If f(x)anxn an-1xn-1 a1x a0
• Then the possible rational roots are

Factors of the last term (a0) over the factors of
the first term (an)
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Example
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Find all real roots
x
y
0
1
Mult. of 2 Touches.
Goes Through
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Find all real roots
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Find all real roots
x
y
0
3
Goes Through ALL
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Find all real roots
13
Find all real roots
x
y
0
-6
All Go Through
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Find all real roots
15
Find all real rootsDo NOT Graph.
NOT Real!
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Find all real roots
17
Find all real rootsDo NOT Graph.
18
Find all real roots
19
Find all real roots
20
Complex Numbers
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Imaginary Unit (i)
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Complex Numbers

• Consists of a real number plus an imaginary
  number
• Looks like a bi
• Can also be called an imaginary number
• If a 0, then its a pure imaginary number

23
Simplify
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Simplify
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Simplify
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Simplify
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Simplify
28
Simplify
29
Simplify