IBMathsDPAnalysis & Approaches (AA)HLRevision Notes1. Number & AlgebraBinomial TheoremExtension of The Binomial Theorem

Extension of The Binomial Theorem (DP IB Analysis & Approaches (AA)): Revision Note

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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

Examiner Tips and Tricks

  • 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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