How to long divide (x^3-2x^2-4x-4)/(x^2+x-2)?

1 Answer
Apr 16, 2017

(x^3-2x^2-4x-4)/(x^2+x-2)=x-3+4/(x+2)+3/(x-1)

Explanation:

(x^3-2x^2-4x-4)/(x^2+x-2)

By long division,

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Hence,

(x^3-2x^2-4x-4)/(x^2+x-2)=x-3+color(green)((x-10)/(x^2+x-2)

Then, let a and b be unknowns,

color(green)((x-10)/(x^2+x-2))=(x-10)/((x+2)(x-1))
color(white)(xxxxxx//x)=a/(x+2)+b/(x-1)

Multiply throughout by x^2+x-2,

x-10=a(x-1)+b(x+2)

When color(red)(x=1,

color(red)(1)-10=a(color(red)(1)-1)+b(color(red)(1)+2)
color(white)(xxx)3b=-9
color(white)(xxx3)b=-3

When color(blue)(x=-2,

color(blue)(-2)-10=a(color(blue)(-2)-1)+b(color(blue)(-2)+2)
color(white)(....)-3a=-12
color(white)(....-3)a=4

Hence, substitute a=4 and b=-3,

(x^3-2x^2-4x-4)/(x^2+x-2)=x-3+4/(x+2)+3/(x-1)