Question

The HCF of two expressions is $h\left(x\right)=(x+3)$ and their LCM is $m\left(x\right)=x^3-7x+6$. If one polynomial is $q\left(x\right)=x^2+x-6$ and the other polynomial $p\left(x\right)$ has negative leading coefficient, find the other polynomial.

Answer 1

$h\left(x\right)=(x+3)$ is HCF of $p\left(x\right)$ and $q\left(x\right)$.

So, both the polynomials will contain $(x+3)$ as one of the factor.

Now, $q\left(x\right)=x^2+x-6=(x+3)(x-2)$ ...... $\left(i\right)$

$m\left(x\right)$ is LCM of $p\left(x\right)$ and $q\left(x\right)$. So, it will also contain $(x+3)$ as one of the factor.

$\therefore m\left(x\right)=x^3-7x+6=(x+3)f\left(x\right)$

We first find $f\left(x\right)$,

By using long division method, we get

$f\left(x\right)=(x^2-3x+2)$

$\therefore m\left(x\right)=(x+3)(x^2-3x+2)=(x+3)(x-2)(x-1)$

Now, $p\left(x\right)$ has $(x+3)$ as one of the factor.

Since, LCM contains $(x-1)$ and $q\left(x\right)$ does not contains $(x-1)$ as one of the factors. So, $p\left(x\right)$ will have $(x-1)$ and it will be another factor.

As it is given that, $p\left(x\right)$ has negative leading coefficient, so we have

$p\left(x\right)=-(x+3)(x-1)=-(x^2+2x-3)=-x^2-2x+3$