If $x + a$ is a common factor of expressions $f (x) = x^{2} + p x + q$ and $g (x) = x^{2} + m x + n$ , then 'a' is equal to:

\frac{m}{n}

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\frac{n + q}{m + p}

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\frac{n - q}{m - p}

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\frac{n}{m}

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Solution

The correct option is C $\frac{n - q}{m - p}$
Given:
$x + a$ is a factor of polynomials $f (x) = x^{2} + p x + q$ and $g (x) = x^{2} + m x + n$

Using factor theorem,
$f (- a) = 0$
$a^{2} - a p + q = 0 (i)$

Again using factor theorem,
$g (- a) = 0$
$g (- a) = a^{2} - a m + n (i i)$

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

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If x + a is a common factor of expressions $f (x) = x^{2} + p x + q a n d g (x) = x^{2} + m x + n;$ show that : $a = \frac{n - q}{m - p}$

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