7.3 Q-2 | Mathematical Induction And Binomial Theorem | Mathematics 11

Exercise 7.3 — Question 2

This question applies the Binomial Theorem to expand binomial expressions and identify specific terms using the general term formula.

For the expansion of $(a + b)^{n}$ where $n \in N$ :

$T_{r + 1} = (r n) a^{n - r} b^{r}, r = 0, 1, 2, \dots, n$

The expansion has $(n + 1)$ terms total.

Using the Binomial Theorem with $a = x$ , $b = 2$ , $n = 4$ :

$(x + 2)^{4} = (0 4) x^{4} + (1 4) x^{3} (2) + (2 4) x^{2} (2)^{2} + (3 4) x (2)^{3} + (4 4) (2)^{4}$

$= x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16$

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

For the term independent of $x$ , set the power of $x$ equal to zero: $6 - 2 r = 0 ⟹ r = 3$

$T_{4} = (3 6) = 20$

General term: $T_{r + 1} = (r 5) (2)^{5 - r} (x)^{r}$

For $x^{3}$ , set $r = 3$ : $T_{4} = (3 5) (2)^{2} = 10 \times 4 = 40$

The coefficient of $x^{3}$ is $40$ .

Task	Method
Expand fully	Apply $T_{r + 1} = (r n) a^{n - r} b^{r}$ for all $r$
Find a specific term	Identify $r$ from the required power
Term independent of $x$	Set power of $x$ in $T_{r + 1}$ equal to $0$
Middle term	Use $r = n /2$ (if $n$ is even)