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

May 29, 2016

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)

May 29, 2016

Same answer, just a different presentation

$\text{ } 2 \left(x - y\right) \left(b + 1\right)$

#### Explanation:

Given:$\text{ } 2 b x + 2 x - 2 b y - 2 y$

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

Grouping

$\text{ } \left(2 b x + 2 x\right) + \left(- 2 b y - 2 y\right)$

Looking for common factors within each bracket

$\text{ } 2 x \left(b + 1\right) - 2 y \left(b + 1\right)$

Notice that $\left(b + 1\right)$ is a common factor

$\text{ } \left(b + 1\right) \left(2 x - 2 y\right)$

Notice that 2 is a common factor in the last bracket

$\text{ } 2 \left(b + 1\right) \left(x - y\right)$

$\textcolor{red}{\text{This is equally true if you reverse the order of the brackets}}$

$\text{ } 2 \left(x - y\right) \left(b + 1\right)$