# How do you write y=2x^3+10x^2+12x in factored form?

Sep 29, 2015

$y = 2 x \left(x + 3\right) \left(x + 2\right)$

#### Explanation:

When factoring, we look for something called a common factor; in other words, something that appears in all of the terms. Right off the bat, we can see that all of our coefficients (the numbers next to the $x$s) are even - so 2 is a common factor. In addition, all of our terms contain an $x$. Putting it all together, our first common factor is $2 x$:

$2 x \left({x}^{2} + 5 x + 6\right)$

If we were to distribute the $2 x$, we would get $2 {x}^{3} + 10 {x}^{2} + 12 x$, which is our original problem.

Now, we proceed to factoring the quadratic equation ${x}^{2} + 5 x + 6$. We do this by looking for two numbers that add to 5 and multiply to 6; these numbers are 3 and 2. If you're wondering why: when we factor a quadratic like this one, we want it in the form $\left(x + a\right) \left(x + b\right)$, where $a$ and $b$ multiply to the constant term and add to the first-degree term (in this case, our constant term is $6$ and our first degree term is $5 x$). Here, we have $a = 3$ and $b = 2$ ($a = 2$ and $b = 3$ works too). So, the factored form of the quadratic equation ${x}^{2} + 5 x + 6 = \left(x + 3\right) \left(x + 2\right)$. Back to the main problem.

Now that we have our quadratic completely factored, we can finish up. We simply replace our quadratic with the factored version, like this:

$2 x \left(x + 3\right) \left(x + 2\right)$

Since we can't simplify this any further, we can say that $y = 2 {x}^{3} + 10 {x}^{2} + 12 x$ in factored form is $y = 2 x \left(x + 3\right) \left(x + 2\right)$.