Question

How can we factorise 1+a+b+c+ab+bc+ca+abc

Answer 1

(1+a)+b(1+a)+c(1+a) +bc(1+a)
or(1+a)(1+b+c+bc)
or(1+a){(1+b)+c(1+b)}
or(1+a)(1+b)(1+c)

Answer 2

Answer:

The factorization of $1+a+b+c+ab+bc+ca+abc$ gives $( 1 + a ) ( 1 + b ) ( 1 + c )$

To factorize:

$1+a+b+c+ab+bc+ca+abc$

Solution:

Given expression is,

$1+a+b+c+ab+bc+ca+abc$

To factorise the terms,  

First, rearranging the expression$1+a+b+c+ab+bc+ca+abc$,  

We get,

$\begin{aligned} = & 1 + c + a + c a + b + b c + a b + a b c \\\\ = & 1 ( 1 + c ) + a ( 1 + c ) + b ( 1 + c ) + a b ( 1 + c ) \\\\ & = ( 1 + c ) ( 1 + a + b + a b ) \\\\ = & ( 1 + c ) \{ 1 ( 1 + a ) + b ( 1 + a ) \} \\\\ & = ( 1 + c ) \{ ( 1 + a ) ( 1 + b ) \} \\\\ & = ( 1 + a ) ( 1 + b ) ( 1 + c ) \end{aligned}$

Thus, the factorisation of $1+a+b+c+ab+bc+ca+abc$ gives out the terms $(1+a)(1+b)(1+c)$