How do you factor $6 x^{3} + 21 x^{2} + 15 x$ $6x^3+21x^2+15x$ ?

Question

2736 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-06-26T18:49:57

$6 x^{3} + 21 x^{2} + 15 x = 3 x (2 x^{2} + 7 x + 5)$ $6x^3+21x^2+15x = 3x(2x^2+7x+5)$

$= 3 x (2 x + 5) (x + 1)$ $=3x(2x+5)(x+1)$

Given $6 x^{3} + 21 x^{2} + 15 x$ $6x^3+21x^2+15x$

First notice that all of the terms are divisible by $3 x$ $3x$ , so separate out that as a factor first:

$6 x^{3} + 21 x^{2} + 15 x = 3 x (2 x^{2} + 7 x + 5)$ $6x^3+21x^2+15x = 3x(2x^2+7x+5)$

Let $f (x) = 2 x^{2} + 7 x + 5$ $f(x) = 2x^2+7x+5$

First notice that $2 - 7 + 5 = 0$ $2-7+5 = 0$ , so $f (- 1) = 0$ $f(-1) = 0$ , so $(x + 1)$ $(x+1)$ is a factor of $f (x)$ $f(x)$

The remaining factor must be $(2 x + 5)$ $(2x+5)$ in order that the coefficient $2$ $2$ of $x^{2}$ $x^2$ and the constant term $5$ $5$ result:

$2 x^{2} + 7 x + 5 = (2 x + 5) (x + 1)$ $2x^2+7x+5 = (2x+5)(x+1)$

Putting it all together:

$6 x^{3} + 21 x^{2} + 15 x = 3 x (2 x + 5) (x + 1)$ $6x^3+21x^2+15x = 3x(2x+5)(x+1)$

How do you factor 6x^3+21x^2+15x 6x3+21x2+15x6x^3+21x^2+15x ?