1.6.2 Extension of The Binomial Theorem

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
- ${(a + b)}^{n} = a^{n} {(1 + \frac{b}{a})}^{n}$
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

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

IB DP Maths: AA HL

1.6.2 Extension of The Binomial Theorem