# How do you factor 36a^3b^2+66a^2b^3-210ab^4?

Apr 11, 2017

$36 {a}^{3} {b}^{2} + 66 {a}^{2} {b}^{3} - 210 a {b}^{4} = \textcolor{g r e e n}{\left(6 a {b}^{2}\right) \left(3 a + 5 b\right) \left(2 a - 7 b\right)}$

#### Explanation:

Extracting the common factors from each term:
color(white)("XXX"){: (ul(36),ul(66),ul(210)," | ",6), (ul(a^3),ul(a^2),ul(a)," | ",a), (ul(b^2),ul(b^3),ul(b^4)," | ",b^2) :}
giving
$\textcolor{w h i t e}{\text{XXX}} \left(6 a {b}^{2}\right) \left(6 {a}^{2} + 11 a b - 35 {b}^{2}\right)$

Looking for integer (we are optimists) coefficients as factors for $\left(6 {a}^{2} + 11 a b - 35 {b}^{2}\right)$

color(white)("XXX"){: (ul("factors of 6"),ul("factors of 35"),ul("difference of cross product")), (1,5,), (ul(6),ul(7),ul(23)), (1,7,), (ul(6),ul(5),ul(37)), (2,5,), (ul(3),ul(7),ul(1)), (2,7,), (ul(3),ul(5),ul(11)) :}
...and we have found a set that gives us the coefficient of the middle term!

It only remains to figure out the positive/negative signs to give $+ 11$

$\left(6 a {b}^{2}\right) \left(2 a - 7 b\right) \left(3 a + 5 b\right)$