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

Mar 22, 2018

$2 {c}^{2} d \left(1 - 2 d\right)$

#### Explanation:

$\implies 2 {c}^{2} d - 4 {c}^{2} {d}^{2}$

Next, we identify terms that are similar:

$\implies \textcolor{\mathmr{and} a n \ge}{2} \textcolor{b l u e}{{c}^{2}} \textcolor{red}{d} - \textcolor{\mathmr{and} a n \ge}{4} \textcolor{b l u e}{{c}^{2}} \textcolor{red}{{d}^{2}}$

Let's start with $\textcolor{\mathmr{and} a n \ge}{\text{orange}}$. 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.

$\implies \textcolor{\mathmr{and} a n \ge}{2} \left(\textcolor{b l u e}{{c}^{2}} \textcolor{red}{d} - \textcolor{\mathmr{and} a n \ge}{2} \textcolor{b l u e}{{c}^{2}} \textcolor{red}{{d}^{2}}\right)$

Now let's look at $\textcolor{b l u e}{\text{blue}}$. We have the same term ${c}^{2}$ on both sides of the minus sign. So we can factor this term out.

$\implies 2 \textcolor{b l u e}{{c}^{2}} \left(\textcolor{red}{d} - 2 \textcolor{red}{{d}^{2}}\right)$

Now the last type of term is $\textcolor{red}{\text{red}}$. 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.

$\implies 2 {c}^{2} \textcolor{red}{d} \left(1 - 2 \textcolor{red}{d}\right)$

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

$\implies 2 {c}^{2} d \left(1 - 2 d\right)$

Mar 22, 2018

$\left(2 {c}^{2} d\right) \left(1 - 2 d\right)$

#### 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)) :}

$2 {c}^{2} d - 4 {x}^{2} {d}^{2}$

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

$\textcolor{w h i t e}{\text{XXX")=color(lime)(""(2c^2d))color(purple)(} \left(1 - 2 d\right)}$