Chapter 6Final Touches
Pg 325 #42-47
Determine whether each binomial is a factor of
Reminds me of a “bank”Limits the ones I have to check!Of course a graph helps too!!!
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x 3 + x 2 −16x −16
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42. x + 2 43. x − 4 44. x +1
45. x −1 46. x − 2 47. x + 4
If “it” goes in evenly, it’s a
If “it” goes in evenly, it’s a
FACTOR
If “it” goes in evenly, it’s a
FACTOR
We know the remainder is 0
Ultimate Goal
• To solve polynomials
• To graph polynomials
• If we can get the polynomial in factored form, then life is pretty easy!
Tool to help find factors
• Long Division– Always works
• Synthetic Division– Works with linear factors• Especially when written in the form (x-a)
Tool to help find factors
• We need a BANK to limit the choices
• RATIONAL ROOT THEOREM• If your polynomial is in standard form and• If your polynomial has integer coefficients• Then the bank consists of
• The possible rational roots are found in the bank
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± factors of constant
± factors of "a"
Example
• List the possible rational roots for
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2x 3 + 8x 2 + 2x −12 = 0
Before we extend this…
BOGO, new perspective
• Irrational Root Theorem– If the is a root, then is also a root– If is a root, then is also a root
• Imaginary Root Theorem– If is a root, then is also a root
• They are called conjugates of each other!
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b
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− b
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a− b
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a+ b
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a+ bi
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a−bi
Examples
• Given the polynomial, find the other roots
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3, 4 − 6, 5
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4i, 7, 0, − 3+ 2i
Intermission
• Stand up and stretch for 1 minute
• Then use your calc to answer pg 318 #49
Back to finding all the roots
• Solve:
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−2(x + 5)(x − 4)(x −1) = 0
Back to finding all the roots
• Solve:
• Solve:
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−2(x + 5)(x − 4)(x −1) = 0
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x 4 − x 2 =12
Back to finding all the roots
• Solve:
• Solve:
• Solve: €
−2(x + 5)(x − 4)(x −1) = 0
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x 4 − x 2 =12
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2x 3 − x 2 + 2x −1= 0
Fundamental Theorem of Algebra
• (Corollary) nth degree polynomials have exactly n roots (including real and imaginary)
• has 3 roots. Period.
• has 4 roots. Period.€
x 3 + 2x 2 − 4x − 6 = 0
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x 4 − 81= 0
All done!
• Solve
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x 3 + 2x 2 − 5x −10 = 0
All done!
• Solve
• Solve €
x 3 + 2x 2 − 5x −10 = 0
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2x 3 +14x 2 + 6 = −13x
Study• Form a study group tonight!• Do 6.5 pg 339 #4, 7, 12, 13, 17, 19, 21, 38• Do 6.6 pg 343 #10-12• Download Geist’s notes (from my website)• Review Solving Quadratics• Do Chap 5 Review pg. 300 #24, 27, 31, 38• Friday test: Chap 6 / part of Chap 5 Retest• Monday: Review for Cumulative• Tuesday: Cumulative 5-6 test• Wed: Prepare for Final and take probability test• Friday: 200 point Final