Question

Find the H.C.F of the expressions $x^3+\left(a+b\right)x^2+\left(ab+1\right)x+b$ and $x^3+2a{x}^2+\left(a^2+1\right)x+a.$

Answer 1

We find factors of both polynomials :

$x^3+(a+b)x^2+(ab+1)x+b$

$\Rightarrow x^3+a{x}^2+b{x}^2+abx+x+b$

$\Rightarrow x^3+a{x}^2+x+b{x}^2+abx+b$

$\Rightarrow x(x^2+ax+1)+b(x^2+ax+1)$

$\Rightarrow (x+b)(x^2+ax+1)$

$x^3+2a{x}^2+a^{2}x+x+a$

$\Rightarrow x^3+a{x}^2+x+a{x}^2+a^{2}x+a$

$\Rightarrow (x^2+ax+1)+a(x^2+ax+1)$

$\Rightarrow (x+a)(x^2+ax+1)$

$\therefore$ HCF of $x^3+(a+b)x^2(ab+1)+b$ and $x^3+2a{x}^2+(a^2+1)x+a$

$=x^2+ax+1$