Question

For two polynomial p(x) and q(x), their H.C.F. h(x) $=$ x-2 and their L.C.M. $m\left(x\right)=x^2-3x+2$, find the other polynomial q(x).

• A. $x^2+x+6$
• B. $x^2-x-6$
• C. $x^2+x-6$
• D. $x^2-x+6$

Answer 1

The correct option is C $x^2+x-6$

H.C.F. $h\left(x\right)=$ (x-2)
L.C.M. $m\left(x\right)$ is $x^3-7x+6$
$=x^3-1-7x+7(x^3-1)-7(x-1)$
$=(x-1)(x^3+x-6)=(x-1)(x+3)(x-2)$
$p\left(x\right)=x^2-3x+2=(x-2)(x-1)$
Now, $p\left(x\right).q\left(x\right)$=$h\left(x\right).m\left(x\right)$
$\therefore q\left(x\right)=\frac{h\left(x\right).m\left(x\right)}{p\left(x\right)}=\frac{(x-2)\times (x-1)(x-2)(x+3)}{(x-2)(x-1)}$
$=(x-2)(x+3)=x^2+x-6$
Thus, The other polynomial $q\left(x\right)=x^2+x-6$