Extension of The Binomial Theorem | DP IB Maths: AA HL Revision Notes 2021

Binomial Theorem: Fractional & Negative Indices

The formula given in the formula booklet for the binomial theorem applies to positive integers only

- ${(a + b)}^{n} = a^{n} + {}^{n}{C_{1}} a^{n - 1} b + \dots + {}^{n}{C_{r}} a^{n - r} b^{r} + \dots + b^{n}$

For negative or fractional powers the expression in the brackets must first be changed such that the value for a is 1
- ${(a + b)}^{n} = a^{n} {(1 + \frac{b}{a})}^{n}$
- ${(a + b)}^{n} = a^{n} (1 + n (\frac{b}{a}) + \frac{n (n - 1)}{2!} {(\frac{b}{a})}^{2} + . . .), n \in ℚ$
- This is given in the formula booklet
If a = 1 and b = x the binomial theorem is simplified to
- ${(1 + x)}^{n} = 1 + n x + \frac{n (n - 1)}{2!} x^{2} + \frac{n (n - 1) (n - 2)}{3!} x^{3} + \dots, n \in ℚ, |x| < 1$
- 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
  - $\frac{1}{{(a + b)}^{n}} = {(a + b)}^{- n}$
- Or written as a surd
  - $\sqrt[n]{{(a + b)}^{m}} = {(a + b)}^{\frac{m}{n}}$
For $n \notin ℕ$ the expansion is infinitely long
- You will usually be asked to find the first three terms
The expansion is only valid for $|x| < 1$
- This means $- 1 < x < 1$
- This is known as the interval of convergence
- For an expansion ${(a + b x)}^{n}$ the interval of convergence would be $- \frac{a}{b} < x < \frac{a}{b}$

The binomial expansion can be used to form an approximation for a value raised to a power
Since $|x| < 1$ 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. ${(1 - x)}^{\frac{1}{2}} = 0 . 96^{\frac{1}{2}}$
- STEP 2: Find the value of x by solving the appropriate equation
  - e.g. $\begin{array}{rcl} 1 - x & = & 0.96 \\ x & = & 0.04 \end{array}$

- STEP 3: Substitute this value of x into the expansion to find the approximation
  - e.g. $1 - \frac{1}{2} (0.04) - \frac{1}{8} {(0.04)}^{2} = 0.9798$
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

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

Consider the binomial expansion of $\frac{1}{\sqrt{9 - 3 x}}$ .

Write down the first three terms.

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

State the interval of convergence for the complete expansion.

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

Use the terms found in part (a) to estimate

\frac{1}{\sqrt{10}}

. Give your answer as a fraction.

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