Question

How do you factor `2c^2d-4c^2d^2`?

Answer 1

`2c^2d(1-2d)`

Explanation:

We start with:

`=>2c^2d-4c^2d^2`

Next, we identify terms that are similar:

`=>color(orange)(2)color(blue)(c^2)color(red)(d)-color(orange)(4)color(blue)(c^2)color(red)(d^2)`

Let's start with `color(orange)"orange"`. We have `2` and `4` on opposite sides of the minus sign. The greatest common factor of these two values is `2`, so that is what we factor out first. This will leave a `2` on the RHS of the minus sign.

`=>color(orange)(2)(color(blue)(c^2)color(red)(d)-color(orange)(2)color(blue)(c^2)color(red)(d^2))`

Now let's look at `color(blue)"blue"`. We have the same term `c^2` on both sides of the minus sign. So we can factor this term out.

`=>2color(blue)(c^2)(color(red)(d)-2color(red)(d^2))`

Now the last type of term is `color(red)"red"`. We have one term with a power of `1` and one term with a power of `2`. With powers, we factor out the lowest power `L`. Any terms with a power higher (say `H`) than the lowest will be leftover with a power equal to `H-L`. Let's factor out `d`. Note! The term on the LHS of the minus sign will become `1`, since there are no other terms left after factoring.

`=>2c^2color(red)(d)(1-2color(red)(d))`

Now that we have touched all of the terms, we are finished. The factored version of the expression is:

`=>2c^2d(1-2d)`

Answer 2

`(2c^2d)(1-2d)`

Explanation:

`{: ("terms:",2c^2d,-,4c^2d^2), ("factors:",1xx2xxcxxcxxd,,1xx2xx2xxcxxcxxd xxd), ("common factors (excluding 1):",color(lime)(color(white)(1xx)2xxcxxcxxd)-,,color(lime)(1xx2color(white)(xx2)xxcxxcxxdcolor(white)( xxd))), ("remaining factors:",color(blue)(1color(white)(xx2xxcxxcxxd)),-,color(magenta)(color(white)(1xx2xx)2color(white)(xxcxxcxxd) xxd)) :}`

`2c^2d-4x^2d^2`

`color(white)("XXX")=color(lime)(""(2xxcxxcxxd)) xx (color(blue)1 - color(magenta)(""(2xxd)))`

`color(white)("XXX")=color(lime)(""(2c^2d))color(purple)(""(1-2d))`