7.4 Q-1 | Mathematical Induction And Binomial Theorem | Mathematics 11

Exercise 7.4 — Question 1: Binomial Theorem Expansions

This exercise focuses on applying the Binomial Theorem to expand binomial expressions of the form $(a + b)^{n}$ where $n$ is a positive integer.

Binomial Theorem Statement

For any real numbers $a$ and $b$ and a positive integer $n$ :

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

Expanded form:

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

where the binomial coefficients are:

$(r n) = \frac{n !}{r ! ( n - r )!}$

Pascal's Triangle

Binomial coefficients can also be read directly from Pascal's Triangle:

$n = 0 : n = 1 : n = 2 : n = 3 : n = 4 : 111134121361141$

Each entry is the sum of the two entries directly above it. The $r$ -th entry in row $n$ gives $(r n)$ .

Worked Examples

Example 1: Expand $(x + 2)^{4}$

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

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

$= x^{4} + 4 \cdot 2 x^{3} + 6 \cdot 4 x^{2} + 4 \cdot 8 x + 16$

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

Example 2: Expand $(2 a - 3 b)^{3}$

Write as $(2 a + (- 3 b))^{3}$ with $n = 3$ :

$= (0 3) (2 a)^{3} + (1 3) (2 a)^{2} (- 3 b) + (2 3) (2 a) (- 3 b)^{2} + (3 3) (- 3 b)^{3}$

$= 8 a^{3} + 3 (4 a^{2}) (- 3 b) + 3 (2 a) (9 b^{2}) + (- 27 b^{3})$

$= 8 a^{3} - 36 a^{2} b + 54 a b^{2} - 27 b^{3}$

Example 3: Expand $(x + \frac{1}{x})^{5}$

With $a = x$ , $b = \frac{1}{x}$ , $n = 5$ :

$= (0 5) x^{5} + (1 5) x^{4} \cdot \frac{1}{x} + (2 5) x^{3} \cdot \frac{1}{x ^{2}} + (3 5) x^{2} \cdot \frac{1}{x ^{3}} + (4 5) x \cdot \frac{1}{x ^{4}} + (5 5) \frac{1}{x ^{5}}$

$= x^{5} + 5 x^{3} + 10 x + \frac{10}{x} + \frac{5}{x ^{3}} + \frac{1}{x ^{5}}$

Key Properties

The expansion of $(a + b)^{n}$ has $n + 1$ terms.
The general term ( $(r + 1)$ -th term) is: $T_{r + 1} = (r n) a^{n - r} b^{r}$
The sum of all binomial coefficients: $\sum_{r = 0}^{n} (r n) = 2^{n}$
The Binomial Theorem applies when $n \in N$ (positive integers).

Exercise 7.4 — Question 1: Binomial Theorem Expansions

This exercise focuses on applying the Binomial Theorem to expand binomial expressions of the form $(a + b)^{n}$ where $n$ is a positive integer.

Binomial Theorem Statement

For any real numbers $a$ and $b$ and a positive integer $n$ :

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

Expanded form:

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

where the binomial coefficients are:

$(r n) = \frac{n !}{r ! ( n - r )!}$

Pascal's Triangle

Binomial coefficients can also be read directly from Pascal's Triangle:

$n = 0 : n = 1 : n = 2 : n = 3 : n = 4 : 111134121361141$

Each entry is the sum of the two entries directly above it. The $r$ -th entry in row $n$ gives $(r n)$ .

Worked Examples

Example 1: Expand $(x + 2)^{4}$

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

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

$= x^{4} + 4 \cdot 2 x^{3} + 6 \cdot 4 x^{2} + 4 \cdot 8 x + 16$

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

Example 2: Expand $(2 a - 3 b)^{3}$

Write as $(2 a + (- 3 b))^{3}$ with $n = 3$ :

$= (0 3) (2 a)^{3} + (1 3) (2 a)^{2} (- 3 b) + (2 3) (2 a) (- 3 b)^{2} + (3 3) (- 3 b)^{3}$

$= 8 a^{3} + 3 (4 a^{2}) (- 3 b) + 3 (2 a) (9 b^{2}) + (- 27 b^{3})$

$= 8 a^{3} - 36 a^{2} b + 54 a b^{2} - 27 b^{3}$

Example 3: Expand $(x + \frac{1}{x})^{5}$

With $a = x$ , $b = \frac{1}{x}$ , $n = 5$ :

$= (0 5) x^{5} + (1 5) x^{4} \cdot \frac{1}{x} + (2 5) x^{3} \cdot \frac{1}{x ^{2}} + (3 5) x^{2} \cdot \frac{1}{x ^{3}} + (4 5) x \cdot \frac{1}{x ^{4}} + (5 5) \frac{1}{x ^{5}}$

$= x^{5} + 5 x^{3} + 10 x + \frac{10}{x} + \frac{5}{x ^{3}} + \frac{1}{x ^{5}}$

Key Properties

The expansion of $(a + b)^{n}$ has $n + 1$ terms.
The general term ( $(r + 1)$ -th term) is: $T_{r + 1} = (r n) a^{n - r} b^{r}$
The sum of all binomial coefficients: $\sum_{r = 0}^{n} (r n) = 2^{n}$
The Binomial Theorem applies when $n \in N$ (positive integers).