Question

How do you factor by grouping ax –ay – bx + by?

Answer 1

Hint: In this problem, we need to take out the common terms by grouping. We have four terms in total. We will group these terms in pairs of two, and then take out the common factor in both the pairs. This grouping method is used in polynomials as well as in solving quadratic equations.

Complete step by step answer:
Now, let’s solve the question.
As we know that polynomials are the expressions having one or more terms. These are classified into 3 types which are: monomial, binomial and trinomial. Monomials are expressions having only one single term. Examples of monomials are: 2x, $4{{y}^{2}}$etc. Second one is a binomial which has only two terms. Examples of binomials are: 2x + 4, $6{{z}^{3}}-4x$. And the third one is trinomial which has three terms. Examples of trinomials are: 3x + 4y + 8, $3{{x}^{2}}+2xy+4z$ etc. in question, we are given four terms. So it is just considered as a polynomial. The terms are already split, all we need to do is grouping the terms into pairs so that common terms can be extracted. Here, 2 pairs have to be formed.
First write the expression.
  $\Rightarrow $ax –ay – bx + by
As we can see, the first two terms can be grouped and ‘a’ can be taken out. Rest 2 terms also be grouped and ‘b’ is taken out.
$\Rightarrow a\left( x-y \right)-b\left( x-y \right)$
As we can see, x – y is the common in both the groups. Just take it out:
$\Rightarrow \left( a-b \right)\left( x-y \right)$
We got the 2 factors after the grouping method.

Note:
While grouping, take care of the rules of integers because it is the most important step. If we do not change the signs, we will not get the common terms and could never reach the final solution. For this question, there is one more way of grouping:
$\Rightarrow $ax –ay – bx + by
We can group alternate terms also:
$\Rightarrow $ax – bx – ay + by
$\Rightarrow x\left( a-b \right)-y\left( a-b \right)$
We will get:
$\Rightarrow \left( a-b \right)\left( x-y \right)$
You can see that we got the same factors.