Exercise 7.1 — Question 11: Binomial Theorem Expansion

Binomial Theorem — Quick Reference

For any positive integer $n$ , the Binomial Theorem states:

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

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

where the binomial coefficient is: $(r n) = \frac{n !}{r ! ( n - r )!}$

Key restriction: In its basic form, the Binomial Theorem applies only when $n$ is a positive integer (natural number), producing exactly $n + 1$ terms.

General ( $(r + 1)$ th) Term

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

Worked Solution — Q11

Expand and simplify using the Binomial Theorem.

Example: Expand $(2 x - 3)^{4}$

Write $a = 2 x$ , $b = - 3$ , $n = 4$ .

$(2 x - 3)^{4} = \sum_{r = 0}^{4} (r 4) (2 x)^{4 - r} (- 3)^{r}$

$r$	$(r 4)$	$(2 x)^{4 - r}$	$(- 3)^{r}$	Term
0	1	$16 x^{4}$	$1$	$16 x^{4}$
1	4	$8 x^{3}$	$- 3$	$- 96 x^{3}$
2	6	$4 x^{2}$	$9$	$216 x^{2}$
3	4	$2 x$	$- 27$	$- 216 x$
4	1	$1$	$81$	$81$

$(2 x - 3)^{4} = 16 x^{4} - 96 x^{3} + 216 x^{2} - 216 x + 81$

Key Techniques for Simplification

Identify $a$ , $b$ , and $n$ before expanding.
Handle negative signs carefully — $(- b)^{r}$ is negative when $r$ is odd.
Use the general term $T_{r + 1}$ to find a specific term without full expansion.
Simplify powers of constants and variables at each step.

Practice

Exercise 7.1 — Question 11: Binomial Theorem Expansion

Binomial Theorem — Quick Reference

For any positive integer $n$ , the Binomial Theorem states:

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

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

where the binomial coefficient is: $(r n) = \frac{n !}{r ! ( n - r )!}$

Key restriction: In its basic form, the Binomial Theorem applies only when $n$ is a positive integer (natural number), producing exactly $n + 1$ terms.

General ( $(r + 1)$ th) Term

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

Worked Solution — Q11

Expand and simplify using the Binomial Theorem.

Example: Expand $(2 x - 3)^{4}$

Write $a = 2 x$ , $b = - 3$ , $n = 4$ .

$(2 x - 3)^{4} = \sum_{r = 0}^{4} (r 4) (2 x)^{4 - r} (- 3)^{r}$

$r$	$(r 4)$	$(2 x)^{4 - r}$	$(- 3)^{r}$	Term
0	1	$16 x^{4}$	$1$	$16 x^{4}$
1	4	$8 x^{3}$	$- 3$	$- 96 x^{3}$
2	6	$4 x^{2}$	$9$	$216 x^{2}$
3	4	$2 x$	$- 27$	$- 216 x$
4	1	$1$	$81$	$81$

$(2 x - 3)^{4} = 16 x^{4} - 96 x^{3} + 216 x^{2} - 216 x + 81$

Key Techniques for Simplification

Identify $a$ , $b$ , and $n$ before expanding.
Handle negative signs carefully — $(- b)^{r}$ is negative when $r$ is odd.
Use the general term $T_{r + 1}$ to find a specific term without full expansion.
Simplify powers of constants and variables at each step.

7.1 Q-11

Exercise 7.1 — Question 11: Binomial Theorem Expansion

Binomial Theorem — Quick Reference

General ( $(r + 1)$ th) Term

Worked Solution — Q11

Example: Expand $(2 x - 3)^{4}$

Key Techniques for Simplification

Practice

7.1 Q-11

Exercise 7.1 — Question 11: Binomial Theorem Expansion

Binomial Theorem — Quick Reference

General ( $(r + 1)$ th) Term

Worked Solution — Q11

Example: Expand $(2 x - 3)^{4}$

Key Techniques for Simplification

Practice