Question

How do you factor by grouping $ac-bc+ad-bd$?

Answer 1

Hint: To use the method of factor by grouping in order to factorise the given polynomial $ac-bc+ad-bd$, we need to make two groups, or two pairs of the terms by combining two-two terms from the four terms present in the given polynomial. So we can group the terms as \[\left( ac-bc \right)+\left( ad-bd \right)\]. Then, we can take the factors $c$ and $d$ outside of the two pairs, since they are common to the first and the second pairs respectively. In doing so, we will obtain the polynomial as \[c\left( a-b \right)+d\left( a-b \right)\]. Finally, on taking the common factor \[\left( a-b \right)\] outside, the polynomial will be factored completely.

Complete step by step solution:
Let us write the polynomial given in the above question as
$\Rightarrow p=ac-bc+ad-bd$
To use the factor by grouping method in order to factor the above polynomial, we have to form two groups of two-two terms. For this we can combine the first two terms and the last two terms in the above polynomial to get
$\Rightarrow p=\left( ac-bc \right)+\left( ad-bd \right)$
Now, since the factors $c$ and $d$ are common to the first and the second group in the above polynomial, we can take these outside of the respective groups to get
$\Rightarrow p=c\left( a-b \right)+d\left( a-b \right)$
Finally, we take the common factor \[\left( a-b \right)\] outside to get
$\Rightarrow p=\left( a-b \right)\left( c+d \right)$
Hence, the given polynomial is completely factored using the factor by grouping method as $\left( a-b \right)\left( c+d \right)$.

Note:
We can combine the first term with the third term, and the second term with the third term in the given polynomial to factor it by grouping. In this case, the grouping will look like $\left( ac+ad \right)+\left( -bc-bd \right)$. In this case, we will take the factors $a$ and $-b$ outside of the groups to factor the polynomial.