Question

If $x+a$ is a common factor of expressions $f\left(x\right)=x^2+px+q$ and $g\left(x\right)=x^2+mx+n$, then 'a' is equal to:

• A. $\frac{n-q}{m-p}$
• B. $\frac{n+q}{m+p}$
• C. $\frac{n}{m}$
• D. $\frac{m}{n}$

Answer 1

The correct option is A $\frac{n-q}{m-p}$

Given:
$x+a$ is a factor of polynomials$f\left(x\right)=x^2+px+q$ and $g\left(x\right)=x^2+mx+n$

Using factor theorem,
$f(-a)=0$
$a^2-ap+q=0\left(i\right)$

Again using factor theorem,
$g(-a)=0$
$g(-a)=a^2-am+n\left(ii\right)$

Equating (i) and (ii), we get:
$a^2-ap+q=a^2-am+n$
$am-ap=n-q$
$a(m-p)=n-q$
$a=\frac{n-q}{m-p}$