(x^3-2x^2-4x-4)/(x^2+x-2)x3−2x2−4x−4x2+x−2
By long division,
![enter image source here]()
Hence,
(x^3-2x^2-4x-4)/(x^2+x-2)=x-3+color(green)((x-10)/(x^2+x-2)x3−2x2−4x−4x2+x−2=x−3+x−10x2+x−2
Then, let aa and bb be unknowns,
color(green)((x-10)/(x^2+x-2))=(x-10)/((x+2)(x-1))x−10x2+x−2=x−10(x+2)(x−1)
color(white)(xxxxxx//x)=a/(x+2)+b/(x-1)×××/x=ax+2+bx−1
Multiply throughout by x^2+x-2x2+x−2,
x-10=a(x-1)+b(x+2)x−10=a(x−1)+b(x+2)
When color(red)(x=1x=1,
color(red)(1)-10=a(color(red)(1)-1)+b(color(red)(1)+2)1−10=a(1−1)+b(1+2)
color(white)(xxx)3b=-9×x3b=−9
color(white)(xxx3)b=-3×x3b=−3
When color(blue)(x=-2x=−2,
color(blue)(-2)-10=a(color(blue)(-2)-1)+b(color(blue)(-2)+2)−2−10=a(−2−1)+b(−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)
