DP IB Maths: AA HL

Revision Notes

1.6.2 Extension of The Binomial Theorem

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Binomial Theorem: Fractional & Negative Indices

How do I use the binomial theorem for fractional and negative indices?

  • The formula given in the formula booklet for the binomial theorem applies to positive integers only
    • open parentheses a plus b close parentheses to the power of n equals a to the power of n plus blank scriptbase straight C subscript 1 end scriptbase presubscript blank presuperscript n a to the power of n minus 1 end exponent b plus horizontal ellipsis plus scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n a to the power of n minus r end exponent b to the power of r plus horizontal ellipsis plus b to the power of n
    • where scriptbase straight C subscript r end scriptbase presubscript blank presuperscript n equals fraction numerator n factorial over denominator r factorial open parentheses n minus r close parentheses factorial end fraction
  • For negative or fractional powers the expression in the brackets must first be changed such that the value for a is 1
    • open parentheses a plus b close parentheses to the power of n equals a to the power of n open parentheses 1 plus b over a blank close parentheses to the power of n 
    • open parentheses a plus b close parentheses to the power of n equals a to the power of n open parentheses 1 plus n open parentheses b over a close parentheses space plus fraction numerator n left parenthesis n minus 1 right parenthesis over denominator 2 factorial end fraction open parentheses b over a close parentheses squared space plus space... space space blank close parentheses space comma space n element of straight rational numbers
    • This is given in the formula booklet 
  • If a = 1 and b = x the binomial theorem is simplified to
    • open parentheses 1 plus x close parentheses to the power of n equals 1 plus n x plus fraction numerator n open parentheses n minus 1 close parentheses over denominator 2 factorial end fraction x squared plus fraction numerator n open parentheses n minus 1 close parentheses open parentheses n minus 2 close parentheses over denominator 3 factorial end fraction x cubed plus horizontal ellipsis comma blank n element of straight rational numbers blank comma double-struck    open vertical bar x close vertical bar less than 1
    • This is not in the formula booklet, you must remember it or be able to derive it from the formula given
  • You need to be able to recognise a negative or fractional power
    • The expression may be on the denominator of a fraction
      • 1 over open parentheses a plus b close parentheses to the power of n equals open parentheses a plus b close parentheses to the power of negative n end exponent
    • Or written as a surd
      • n-th root of open parentheses a plus b close parentheses to the power of m end root equals open parentheses a plus b close parentheses to the power of m over n end exponent
  • For n blank not an element of blank straight natural numbers the expansion is infinitely long
    • You will usually be asked to find the first three terms
  • The expansion is only valid for open vertical bar x close vertical bar less than 1
    • This means negative 1 less than x less than 1
    • This is known as the interval of convergence
    • For an expansion open parentheses a blank plus blank b x close parentheses to the power of n the interval of convergence would be negative a over b less than x less than a over b

How do we use the binomial theorem to estimate a value?

  • The binomial expansion can be used to form an approximation for a value raised to a power
  • Since open vertical bar x close vertical bar less than 1 higher powers of x will be very small
    • Usually only the first three or four terms are needed to form an approximation
    • The more terms used the closer the approximation is to the true value
  • The following steps may help you use the binomial expansion to approximate a value
    • STEP 1: Compare the value you are approximating to the expression being expanded
      • e.g.  left parenthesis 1 space minus space x right parenthesis to the power of 1 half end exponent space equals space 0.96 to the power of 1 half end exponent
    • STEP 2: Find the value of x by solving the appropriate equation
      • e.g.  table attributes columnalign right center left columnspacing 0px end attributes row blank blank blank row cell 1 space minus space x space end cell equals cell space 0.96 end cell row cell space x space end cell equals cell space 0.04 end cell end table
    • STEP 3: Substitute this value of x into the expansion to find the approximation
      • e.g.  1 minus 1 half open parentheses 0.04 close parentheses minus 1 over 8 blank open parentheses 0.04 close parentheses squared equals 0.9798
  • Check that the value of x is within the interval of convergence for the expression
    • If x is outside the interval of convergence then the approximation may not be valid

Exam Tip

  • Students often struggle with the extension of the binomial theorem questions in the exam, however the formula is given in the formula booklet 
    • Make sure you can locate the formula easily and practice substituting values in
    • Mistakes are often made with negative numbers or by forgetting to use brackets properly
      • Writing one term per line can help with both of these

Worked example

Consider the binomial expansion of  fraction numerator 1 over denominator square root of 9 blank minus blank 3 x end root end fraction.

a)
Write down the first three terms.

1-6-2-ib-hl-aa-ext-bin-theorem-we-a

b)
State the interval of convergence for the complete expansion.

1-6-2-ib-hl-aa-ext-bin-theorem-we-b

c)
Use the terms found in part (a) to estimate begin mathsize 14px style fraction numerator 1 over denominator square root of 10 end fraction end style . Give your answer as a fraction.

1-6-2-ib-hl-aa-ext-bin-theorem-we-c

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Author: Amber

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.