Question

The HCF of two polynomials is $h\left(a\right)=a-7$ and their LCM is $m\left(a\right)=a^3-10a^2+11a+70$. If one of the polynomial is $p\left(a\right)=a^2-12a+35$, and if the leading coefficient of $q\left(a\right)$ is positive, find the other polynomial $q\left(a\right)$.

Answer 1

Given : $h\left(a\right)=a-7$ and LCM $m\left(a\right)=a^3-10a^2+11a+70$.

$p\left(a\right)=a^2-12a+35$

$=a^2-7a-5a+35$

$=a(a-7)-5(a-7)$

$=(a-7)(a-5)$

$\therefore p\left(a\right)=(a-7)(a-5)$ ......... $\left(i\right)$

Now, $m\left(a\right)=a^3-10a^2+11a+70$

$=(a-7)(a^2-3a-10)$ ........ [Using synthetic division]

$=(a-7)(a^2-5a+2a-10)$

$=(a-7)(a-5)(a+2)$

$\therefore m\left(a\right)=(a-7)(a-5)(a+2)$ ........ $\left(ii\right)$

$q\left(a\right)$ will have $(a-7)$ as one of the factor, since $h\left(a\right)=a-7$

From $\left(ii\right)$ and $\left(i\right)$, $q\left(a\right)$ will have $(a+2)$ as one of the factor.

Also, since $h\left(a\right)\cdot m\left(a\right)=p\left(a\right)\cdot q\left(a\right)$

$⟹q\left(a\right)=(a-7)(a+2)$

Hence, $q\left(a\right)=a^2-5a-14$