Mathematics for AVT Dr Ian Drumm Week 6

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Mathematics for AVTDr Ian DrummWeek 6

• Aims
• Partial Fractions
• Objectives
• Polynomials - factorising
• Equating Coefficients, Proper and Improper
  Fractions,
• Partial Fractions

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Polynomials

• Equations of the form
• For example
• Must be non-negative whole number powers
• Degree of polynomial is highest power
• Cubic polynomial has a degree of 3

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Quadratic Equations

• A quadratic equation is and equation of the form
• Solving finding roots
• Factorisation
• Completing the Square
• Using a formula

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Factorisation split into factors to find roots
(remember multiply by first coefficient and hence
express using z4x)
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Completing the square

• It can be shown that
• For examplecomplete square of
• We know k8 hence
• Hence roots

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Solution by formula

• Given quadratic equation of the form
• Solutions of the form

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Types of roots
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Factorising cubic equations Equating coefficients

• Factorise
• given a factor is
• Write in terms of a, b and c hence find unknowns

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Verifying Solution

• Substitute for x
• If we know x4 is a root, we can write polynomial
  as

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Solving polynomial graphically

• Roots where y0
• Use computer graphing package (e.g. matlab,
  excel, etc.)

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Partial Fractions

• Splitting complicated fractions into simpler
  ones, particularly useful for integration
• For example

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Proper and improper fractions

• Degree of numerator n is highest power in the
  numerator
• Degree of denominator d is highest power in
  denominator
• Proper if dgtn, otherwise improper if d n
• For example n3gtd2 hence improper

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Linear factors

• A linear factor in the
  denominator gives rise to a single partial
  fraction
• For example the expression here has two linear
  factors to give two partial fractions
• Hence we can find the unknowns A and B

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Find partial fractions(proper fraction)

• Split denominator into linear factors
• Multiply partial fractions by linear factors
• Choose value of x to eliminate a term, e.g. x-1
• Hence can rearrange and compare coefficients to
  find A

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Proper Fractions with repeated linear factors

• A repeated linear factor
• in the denominator produces two partial
  fractions
• For example

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Proper fractions with quadratic factors

• Where your denominator cant be factorised.
• A quadratic factor in
  denominator gives partial fraction
• For example

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Find partial fractionImproper Fraction

• Extra polynomial added to partial fractions that
  arise
• Subtract degree of numerator from degree of
  denominator, hence polynomial to degree of n-d
• For examplen2 d1 hence add