Question 5

We will use the distributive property to expand each expression. This means we multiply each term in the first bracket by every term in the second bracket.

Step 1 — Expand the first expression

Let us expand the expression $(a - b) (a + b)$.

We multiply $a$ by $(a+b)$ and then multiply $-b$ by $(a+b)$.

$(a - b) (a + b) = a(a + b) - b(a + b)$

$= a \times a + a \times b - b \times a - b \times b$

$= a^2 + ab - ab - b^2$

$= a^2 - b^2$

$\boxed{a^2 - b^2}$

Step 2 — Expand the second expression

Now, let us expand the expression $(a - b) (a^2 + ab + b^2)$.

We multiply $a$ by $(a^2 + ab + b^2)$ and then multiply $-b$ by $(a^2 + ab + b^2)$.

$(a - b) (a^2 + ab + b^2) = a(a^2 + ab + b^2) - b(a^2 + ab + b^2)$

$= (a \times a^2 + a \times ab + a \times b^2) - (b \times a^2 + b \times ab + b \times b^2)$

$= (a^3 + a^2b + ab^2) - (a^2b + ab^2 + b^3)$

$= a^3 + a^2b + ab^2 - a^2b - ab^2 - b^3$

$= a^3 - b^3$

$\boxed{a^3 - b^3}$

Step 3 — Expand the third expression

Next, we expand the expression $(a - b)(a^3 + a^2b + ab^2 + b^3)$.

We multiply $a$ by $(a^3 + a^2b + ab^2 + b^3)$ and then multiply $-b$ by $(a^3 + a^2b + ab^2 + b^3)$.

$(a - b)(a^3 + a^2b + ab^2 + b^3) = a(a^3 + a^2b + ab^2 + b^3) - b(a^3 + a^2b + ab^2 + b^3)$

$= (a \times a^3 + a \times a^2b + a \times ab^2 + a \times b^3) - (b \times a^3 + b \times a^2b + b \times ab^2 + b \times b^3)$

$= (a^4 + a^3b + a^2b^2 + ab^3) - (a^3b + a^2b^2 + ab^3 + b^4)$

$= a^4 + a^3b + a^2b^2 + ab^3 - a^3b - a^2b^2 - ab^3 - b^4$

$= a^4 - b^4$

$\boxed{a^4 - b^4}$

Step 4 — Identify the pattern

Let us look at the results we found:

1. $(a - b) (a + b) = a^2 - b^2$
2. $(a - b) (a^2 + ab + b^2) = a^3 - b^3$
3. $(a - b)(a^3 + a^2b + ab^2 + b^3) = a^4 - b^4$

We can see a clear pattern here. In each case, we are multiplying $(a-b)$ by a special sum of terms. The result is always of the form $a^n - b^n$. The power 'n' in the result is one more than the highest power of 'a' in the second bracket.

The second bracket always contains terms where the power of 'a' decreases from $n-1$ down to $0$, and the power of 'b' increases from $0$ up to $n-1$.

Step 5 — State the next identity

Following this pattern, the next identity would be for $n=5$.

The second bracket will start with $a^{5-1} = a^4$. The powers of 'a' will decrease, and powers of 'b' will increase. So, the second bracket will be $(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$.

The next identity in the pattern is: $(a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4) = a^5 - b^5$

Step 6 — Check the next identity by expanding

Let us expand the expression $(a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$ to check our pattern.

We multiply $a$ by the second bracket and then multiply $-b$ by the second bracket.

$(a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4) = a(a^4 + a^3b + a^2b^2 + ab^3 + b^4) - b(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$

$= (a^5 + a^4b + a^3b^2 + a^2b^3 + ab^4) - (a^4b + a^3b^2 + a^2b^3 + ab^4 + b^5)$

$= a^5 + a^4b + a^3b^2 + a^2b^3 + ab^4 - a^4b - a^3b^2 - a^2b^3 - ab^4 - b^5$

$= a^5 - b^5$

$\boxed{a^5 - b^5}$ Our expansion matches the pattern.

Answer

(i) $(a - b) (a + b) = a^2 - b^2$ (ii) $(a - b) (a^2 + ab + b^2) = a^3 - b^3$ (iii) $(a - b)(a^3 + a^2b + ab^2 + b^3) = a^4 - b^4$

Yes, there is a pattern. When we multiply $(a-b)$ by a sum of terms where the power of 'a' decreases and the power of 'b' increases, the result is $a^n - b^n$.

The next identity in the pattern is: $(a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4) = a^5 - b^5$

Checking by expanding confirms this result.