How do you factor $2 x^{3} + 4 x^{2} - 8 x$ $2x^3 + 4x^2 - 8x$ ?

Question

How do you factor $2 x^{3} + 4 x^{2} - 8 x$ $2x^3 + 4x^2 - 8x$ ?

2 Answers

Impact of this question

4429 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-05-16T19:44:18

First, we must identify the common elements in this function. Let's just disaggregate them for a while:

$2 \cdot x \cdot x \cdot x + 2 \cdot 2 \cdot x \cdot x - 2 \cdot 2 \cdot 2 \cdot x$ $color(green)(2*x)*x*x+2*color(green)(2*x)*x-2*2*color(green)(2*x)$

Note that the three of them are multiplied by $2 x$ $2x$ . Therefore, this element comes out:

$2 x (x^{2} + 2 x - 4)$ $2x(x^2+2x-4)$

As inside the parenthesis we have a quadratic, we can solve it and find its roots, which are $x_{1} = - 1 - \sqrt{5}$ $x_1=-1-sqrt(5)$ and $x_{2} = - 1 + \sqrt{5}$ $x_2=-1+sqrt(5)$ .

We can rewrite these roots as

$x_{1} + 1 + \sqrt{5} = 0$ $x_1+1+sqrt(5)=0$ and $x_{2} + 1 - \sqrt{5} = 0$ $x_2+1-sqrt(5)=0$

...and use these as our factors:

$2 x (x + (1 + \sqrt{5})) (x + (1 - \sqrt{5}))$ $2x(x+(1+sqrt(5)))(x+(1-sqrt(5)))$

Answer 2 · 2015-05-16T19:45:51

$2 x^{3} + 4 x^{2} - 8 x = 2 x (x^{2} + 2 x - 4)$ $2x^3+4x^2-8x = 2x(x^2+2x-4)$

$x^{2} + 2 x - 4$ $x^2+2x-4$ is of the form $a x^{2} + b x + c$ $ax^2+bx+c$ with $a = 1$ $a=1$ , $b = 2$ $b=2$ and $c = - 4$ $c=-4$ .

The discriminant of this quadratic is

$Delta = b^2-4ac = 2^2-4xx1xx-4 = 4+16 = 20$

Since this is positive, the quadratic equation $x^2+2x-4 = 0$ has 2 distinct real roots. Unfortunately, $20$ is not a perfect square, so those roots are not rational.

The roots of $x^2+2x-4 = 0$ are:

$x = (-b+-sqrt(Delta))/(2a)$

$=(-2+-sqrt(20))/2$

$=(-2+-2sqrt(5))/2$

$=-1+-sqrt(5)$

So $(x - (-1+sqrt(5))) = (x+1-sqrt(5))$

and $(x - (-1-sqrt(5))) = (x+1+sqrt(5))$ are both factors.

$x^2+2x-4 = (x+1-sqrt(5))(x+1+sqrt(5))$

So if we are allowed irrational factors,

$2x^3+4x^2-8x = 2x(x+1-sqrt(5))(x+1+sqrt(5))$

Otherwise, we have to stop at

$2x^3+4x^2-8x = 2x(x^2+2x-4)$

Science

Math

Humanities

... and beyond

How do you factor $2 x^{3} + 4 x^{2} - 8 x$ $2x^3 + 4x^2 - 8x$ ?

2 Answers

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you factor 2x^3 + 4x^2 - 8x2x3+4x2−8x2x^3 + 4x^2 - 8x?

2 Answers

Related questions

Impact of this question

How do you factor $2 x^{3} + 4 x^{2} - 8 x$ $2x^3 + 4x^2 - 8x$ ?