Question

How do you factor completely `2bx+2x-2by-2y`?

Answer 1

2(b+1)(x-y)

Explanation:

There is a common factor of 2 in all 4 terms which can be taken out.

hence 2 (bx + x - by -y).............(A)

now take out the common factor of x in the first 2 terms inside bracket.

thus bx + x = x(b +1)

Similarly a common factor of -y can be removed from the 2 remaining terms.

thus -by - y = -y(b + 1)

now bx+x-by-y = x(b+1)-y(b + 1)

There is a common factor of (b +1) here.

hence: (b +1)(x -y) when factorised.

Putting this back into (A) gives.

2bx+2x-2by-2y = 2(b+1)(x-y)

Answer 2

Same answer, just a different presentation

`" "2(x-y)(b+1)`

Explanation:

Given:`" "2bx+2x-2by-2y`

Sometimes these are spotted straight away, other times not and need a bit of work.

Grouping

`" "(2bx+2x)+(-2by-2y)`

Looking for common factors within each bracket

`" "2x(b+1)-2y(b+1)`

Notice that `(b+1)` is a common factor

`" "(b+1)(2x-2y)`

Notice that 2 is a common factor in the last bracket

`" "2(b+1)(x-y)`

`color(red)("This is equally true if you reverse the order of the brackets")`

`" "2(x-y)(b+1)`