Rational Equations and Partial Fractions

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Rational Equations and Partial Fractions

• Section 4-6
• Pages 243-250

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Advice for success from the Idiots Guide

• Make sure to understand what the major vocabulary
  words mean
• Sift thru the complicated wording and theorems to
  the heart of the meaning
• Develop a mathematical instinct
• Sometimes you just have to memorize

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Objectives

• Review previous learning
• Solve rational equations and inequalities
• Decompose a fraction into partial fractions

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So what am I doing here

• Chapter 4 is trying to find a variety of methods
  of working with polynomials.
• Polynomials are just junk added/subtracted from
  one another. This junk is just numbers and
  variables.
• This junk is used later to describe real-life
  situations.

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What do I know up until this point?

• Every polynomial of degree n can be written as a
  product of k and n linear factors.
• Ex If P(x) has roots 2,4,5,-6, then I can write
  (x-2)(x-4)(x-5)(x6) 0
• If n is a root, then P(n) 0
• Degree of polynomial determines the exact number
  of complex roots.
• Degree also tells me the max. of times graph
  crosses x-axis.
• Odd degrees must cross x-axis at least once.
• Even degrees dont have to cross x-axis

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What do I know up until this point?

• 3. Methods of solving quadratics
• Graphing
• Factoring
• Completing the square
• Quadratic Formulas
• Determinant tells me the nature of the roots
• 4. Remainder Theorem says if I divide the
  polynomial by x-r, where r is a factor, then its
  remainder must be 0

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What do I know up until this point?

• 5. Rational Root Theorem says we look at the
  factors of an and a0 to determine roots. Example
• Factors of an we call p, factors of a0 we call q.
• If p,q have no common factors, then possible root
  
• 6. Rule of Signs for positive roots look at the
  of sign changes. For negative roots take P(-x)
  then look at the sign changes.
• 7. Both rational root and rule of signs narrow
  down our choices of roots. NOTE, once we narrow
  down we use synthetic division to find actual
  roots or plug and chug back into P(x)

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What do I know up until this point?

• 8. Locating zeros thru approximation is easy and
  we call it the Location Principle.
• Use Rational Root and Rule of signs to narrow
  down the roots. OR Graph it.
• Use synthetic division and look for zeros and
  change of signs in remainder.
• Use Calculator table to find sign changes.
• This gives us a good range.

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What do I know up until this point?

• 10. Upper and Lower Bounds confirm whether you
  found all the real zeros
• Upper bounds Take possible positives integer
  roots (rational root theorem) and synthetically
  divide into polynomial. If there is NO sign
  changes (rule of signs have all nos) then it is
  an upper bound.
• Lower bounds Change P(x) to P(-x). Then Take
  possible roots and synthetically divide into the
  new polynomial. If there is NO sign change then
  the negative of the root is the lower bound.
• And there you have it!

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Rational Equations- not really new

• These are those pesky equations with xs on the
  top and bottom of fractions.- So just multiply
  entire equation by the Least Common Denominator.

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Decompose Partial Fractions

• Factor denominator
• Express as sum of 2 fractions using A and B
• Mult. By LCD
• Get rid of denominator

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• 5. Eliminate A
• 6. Eliminate B

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A 2, B 1

• Put it back in

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Rational Inequalities
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x 31 Now must test
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Assignment

• Page 247 13 33 odd