Question

How do you factor `5x^4-40x+10x^3-20x^2`?

Answer 1

`5x(x-2)(x+2)^2`

Explanation:

`"take out a "color(blue)"common factor "5x`

`5x(x^3-8+2x^2-4x)`

`"note that "2^3-8+2(2)^2-4(2)=0`

`rArr(x-2)" is a factor of "x^3+2x^2-4x-8`

`"divide "x^3+2x^2-4x-8" by "(x-2)`

`rArrcolor(red)(x^2)(x-2)color(magenta)(+2x^2)+2x^2-4x-8`

`=color(red)(x^2)(x-2)color(red)(+4x)(x-2)color(magenta)(+8x)-4x-8`

`=color(red)(x^2)(x-2)color(red)(+4x)(x-2)color(red)(+4)(x-2)cancel(color(magenta)(+8))cancel(-8)`

`rArrx^3+2x^2-4x-8=(x-2)(color(red)(x^2+4x+4))`

`color(white)(xxxxxxxxxxxxxxx)=(x-2)(x+2)^2`

`rArr5x^4-40x+10x^3-20x^2`

`=5x(x-2)(x+2)^2`