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