Question

Factorise the following: $ {{p}^{2}}q-p{{r}^{2}}-pq+{{r}^{2}} $

Answer 1

Hint: Factorising an expression polynomial expression is writing the expression in the form of product of the factors of the expression. Try to write the given expression as the product of two terms by taking out the common terms (associativity).

Complete step-by-step answer:
Let us first understand what factorisation is.
Factorising an expression polynomial expression is writing the expression in the form of a product of the factors of the expression.
A factor of an expression is a number or a polynomial expression that exactly divides the expression. This means that when the expression is divided by the factor, the remainder is zero.
Suppose we have numbers 6. We can write the number 6 as $ 6=2\times 3 $ . This means that 2 and 3 are the factors of 6. In other words, 2 and 3 exactly divide the number 6.
Now, to factorise the given polynomial expression $ {{p}^{2}}q-p{{r}^{2}}-pq+{{r}^{2}} $ , let us try to write the expression in the form of product of two expressions.
We can write the given expression as $ {{p}^{2}}q-p{{r}^{2}}-pq+{{r}^{2}}=\left( {{p}^{2}}q-p{{r}^{2}} \right)-\left( pq-{{r}^{2}} \right) $ .
Now, we can see that ‘p’ is common between the two terms in the first bracket. Therefore, take it out of the bracket as a common term.
  $ \Rightarrow {{p}^{2}}q-p{{r}^{2}}-pq+{{r}^{2}}=p\left( pq-{{r}^{2}} \right)-\left( pq-{{r}^{2}} \right) $ .
Now, the term $ \left( pq-{{r}^{2}} \right) $ is common. Therefore, the expression can be written as
 $ \Rightarrow {{p}^{2}}q-p{{r}^{2}}-pq+{{r}^{2}}=\left( pq-{{r}^{2}} \right)\left( p-1 \right) $ .
With this, we have the factored the given expression.
So, the correct answer is “$\left( pq-{{r}^{2}} \right)\left( p-1 \right) $”.

Note: If we want to verify whether the expressions $ \left( pq-{{r}^{2}} \right) $ and $ \left( p-1 \right) $ are the factors of the given expression then we can just divide the one of the expression [say $ \left( p-1 \right) $ ] and check whether it remainder is zero.
The way to factorise an expression is by knowing any one of the factors. We can divide the expression by that factor and find the other factors.