Factorise : 1 + a + b + c + ab + bc + ca + abc
To factorize the expression 1 + a + b + c + ab + bc + ca + abc, we will look for common factors and try to rearrange the terms in a way that allows us to factor out a common expression.
Start by grouping the terms in pairs:
(1 + a) + (b + c) + (ab + bc) + ca + abc
Now, let's factor out the common factors from each pair of terms:
1(1 + a) + 1(b + c) + ab(b + c) + ca + abc
Next, notice that (1 + a) appears twice and (b + c) appears twice. We can factor them out too:
(1 + a)(1 + b + c) + (b + c)(ab + ca) + ca + abc
Now we can see that we have a common factor of (1 + b + c):
(1 + b + c)[ (1 + a) + (b + c)(ab + ca) + ca + abc]
So, the fully factorized expression is:
(1 + b + c) [ (1 + a) + (b + c)(ab + ca) + ca + abc ]