Question

Simplify the following expression:
$(ab+bc)(ab-bc)+(bc+ca)(bc-ac)+(ca+ab)(ac-ab)$$\left[3Marks\right]$

Answer 1

Given expression,
$(ab+bc)(ab-bc)+(bc+ca)(bc-ac)+(ca+ab)(ac-ab)$

Using the identity,
$(a+b)(a-b)=a^2-b^2,$$\left[1Mark\right]$

the given expression can be simplified as
$(ab{)}^2-(bc{)}^2+(bc{)}^2-(ac{)}^2+(ac{)}^2-(ab{)}^2$$\left[1Mark\right]$

By grouping the like terms, this can be further simplified as follows:

$\left[\right(ab{)}^2-\left(ab{)}^2\right]+\left[\right(bc{)}^2-\left(bc{)}^2\right]+\left[\right(ac{)}^2-\left(ac{)}^2\right]=0$$\left[1Mark\right]$