# How do you factor 42x^3 + 51x^2 – 18x?

Jan 31, 2016

$42 {x}^{3} + 51 {x}^{2} - 18 x = \textcolor{g r e e n}{3 x \left(2 x + 3\right) \left(7 x - 2\right)}$

#### Explanation:

Extracting the obvious common factor of $3 x$ from $42 {x}^{3} + 51 {x}^{2} - 18 x$ gives
$\textcolor{w h i t e}{\text{XXX}} 3 x \left(14 {x}^{2} + 17 x - 6\right)$

There are several ways of factoring $\left(14 {x}^{2} + 17 x - 6\right)$

One way is to look for factors $\left(\textcolor{b l u e}{a} \cdot \textcolor{g r e e n}{b}\right)$of $14$ and $\left(\textcolor{c y a n}{c} \cdot \textcolor{\mathmr{and} a n \ge}{d}\right)$ of $\left(- 6\right)$
such that $\textcolor{b l u e}{a} \textcolor{\mathmr{and} a n \ge}{d} - \textcolor{g r e e n}{b} \textcolor{c y a n}{c} = 17$
in order to generate the factors $\left(\textcolor{b l u e}{a} x + \textcolor{c y a n}{c}\right) \left(\textcolor{g r e e n}{b} x + \textcolor{\mathmr{and} a n \ge}{d}\right)$

Table of $\textcolor{b l u e}{a} \textcolor{\mathmr{and} a n \ge}{d} - \textcolor{g r e e n}{b} \textcolor{c y a n}{c}$

{: ("factors of " 14 (axxb)":"," | ",(color(blue)(1) * color(green)(14))," | ",(color(blue)(2) * color(green)(7))," | ",(color(blue)(7) * color(green)(2))," | ",(color(blue)(14) * color(green)(1))), ("----------------------",,"--------------------",,"--------------------",,"--------------------",,"--------------------"), ("factors of "(-6) (cxxd)":"," | ", ," | ", ," | ",," | ", ), (color(white)("XX")((color(cyan)(-1)) * color(orange)(6))," | ",-14+6=-8," | ", -7+12=5," | ",-2+42=40," | ",-1+84=83), (color(white)("XX")((color(cyan)(-2)) * color(orange)(3))," | ", -28+3=-25," | ",-14+6=-8," | ",color(red)(-4+21=17)," | ", ". . ." ), (color(white)("XX")(color(cyan)(2) * (color(orange)(-3)) )," | ",". . ." ," | ", ". . ."," | ",". . ." ," | ", ". . ." ), (color(white)("XX")(color(cyan)(1) * (color(orange)(-6))) ," | ", ". . ." , " | ", ". . .", " | ",". . ." ," | ", ". . .") :}

Which gives us the factoring $\left(a x + c\right) \left(b x + d\right)$
$\textcolor{w h i t e}{\text{XXX}} 14 {x}^{2} + 17 x - 6 = \left(\textcolor{b l u e}{7} x \textcolor{c y a n}{- 2}\right) \left(\textcolor{g r e e n}{2} x \textcolor{\mathmr{and} a n \ge}{+ 3}\right)$

$\Rightarrow 42 {x}^{3} + 52 {x}^{2} - 18 x = \left(3 x\right) \left(7 x - 2\right) \left(2 x + 3\right)$