# How do you factor x^3+7x^2+14x+8?

Mar 5, 2018

color(magenta)(=(x+1)(x+4)(x+2)

#### Explanation:

${x}^{3} + 7 {x}^{2} + 14 x + 8$

$= {x}^{3} - x + 7 {x}^{2} + 15 x + 8$ [Adding and subtracting $x$]

$= x \left({x}^{2} - 1\right) + 7 {x}^{2} + 15 x + 8$

Identity:
color(red)(a^2-b^2=(a+b)(a-b)  &
color(red)((x+a)(x+b)=x^2+(a+b)x+ab

= x(x+1)(x-1)+[7x(x+1)+8(x+1)

$= x \left(x + 1\right) \left(x - 1\right) + \left(x + 1\right) \left(7 x + 8\right)$

$= \left(x + 1\right) \left[x \left(x - 1\right) \left(7 x + 8\right)\right]$

$= \left(x + 1\right) \left[{x}^{2} - x + 7 x + 8\right]$

$= \left(x + 1\right) \left[{x}^{2} + 6 x + 8\right]$

Identity:
color(red)((x+a)(x+b)=x^2+(a+b)x+ab

color(magenta)(=(x+1)(x+4)(x+2)

~Hope this helps! :)

Mar 5, 2018

(x + 1)(x + 2)(x + 4)

#### Explanation:

$f \left(x\right) = {x}^{3} + 7 {x}^{2} + 14 x + 8$
First, we note that f(-1) = - 1 + 7 - 14 + 8 = 0.
So, one factor is (x - 1).
After division, or guest -->
$f \left(x\right) = \left(x + 1\right) \left({x}^{2} + 6 x + 8\right)$
Find 2 numbers knowing the sum (6) and the product (8). They are 2 and 4. Finally,
f(x) = (x + 1)(x + 2)(x + 4)