7.1 Q-30 | Mathematical Induction And Binomial Theorem | Mathematics 11

Exercise 7.1 — Question 30

Expand using the Binomial Theorem and simplify:

$(x - \frac{1}{x})^{6}$

Using the Binomial Theorem, for any positive integer $n$ :

$(a + b)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} b^{r}$

Here $a = x$ , $b = - \frac{1}{x}$ , and $n = 6$ .

The general term is:

$T_{r + 1} = (r 6) (x)^{6 - r} (- \frac{1}{x})^{r} = (r 6) (- 1)^{r} \cdot x^{6 - r} \cdot x^{- r} = (r 6) (- 1)^{r} \cdot x^{6 - 2 r}$

Now expand term by term for $r = 0, 1, 2, 3, 4, 5, 6$ :

$r$	$(r 6)$	$(- 1)^{r}$	$x^{6 - 2 r}$	Term
0	1	$+ 1$	$x^{6}$	$x^{6}$
1	6	$- 1$	$x^{4}$	$- 6 x^{4}$
2	15	$+ 1$	$x^{2}$	$15 x^{2}$
3	20	$- 1$	$x^{0}$	$- 20$
4	15	$+ 1$	$x^{- 2}$	$\frac{15}{x ^{2}}$
5	6	$- 1$	$x^{- 4}$	$- \frac{6}{x ^{4}}$
6	1	$+ 1$	$x^{- 6}$	$\frac{1}{x ^{6}}$

$(x - \frac{1}{x})^{6} = x^{6} - 6 x^{4} + 15 x^{2} - 20 + \frac{15}{x ^{2}} - \frac{6}{x ^{4}} + \frac{1}{x ^{6}}$

Binomial Theorem: $(a + b)^{n} = r = 0 \sum n (r n) a^{n - r} b^{r}$ , valid for positive integer $n$ .
General Term: $T_{r + 1} = (r n) a^{n - r} b^{r}$
When $b$ is negative, alternate signs appear via $(- 1)^{r}$ .
Simplification involves combining powers of $x$ using index laws.