Question

Given that\(\frac{x^3+x^2-4x-4}{x^3+8x^2+17x+10}=\frac{x+a}{x+b}\) .Evaluate \(a+b\)

Answer 1

\(\frac{x^3+x^2-4x-4}{x^3+8x^2+17x+10}=\frac{x^2\left(x+1\right)-4\left(x+1\right)}{x^2\left(x+1\right)+7x\left(x+1\right)+10\left(x+1\right)}\)

\(=\frac{\left(x+1\right)\left(x^2-4\right)}{\left(x+1\right)\left(x^2+7x+10\right)}=\frac{\left(x+2\right)\left(x-2\right)}{x\left(x+2\right)+5\left(x+2\right)}\)

\(=\frac{\left(x+2\right)\left(x-2\right)}{\left(x+2\right)\left(x+5\right)}=\frac{x-2}{x+5}\Rightarrow a=-2;b=5\)

\(\Rightarrow\)\(a+b=-2+5=3\)