Question

if (x+a) is a factor of the polynomial
$x {}^{2} + px + q \: and \: x {}^{2} + mx + n$
prove that
$a = \frac{n - q}{m - p}$

please give full answer​

Answer 1

Answer:

$\sf \red{Given :-}$

(x+a) is a factor of x^2 +px+q and x^2+mx+n

then using the factor theorem which says that the polynomial f(x0 has a factor (x−k) if and only if f(k)=0

We have

$\sf \red{(−a)^2 +p(−a)+q=0 \: \: \: \: \: \: \: \: \: ⟶(1)}$

$\sf \pink{⇒a^2 −ap+q=0} \: \: \: \: \: \: \: \: \: ⟶(2)$

and

$\sf(−a)^2 +m(−a)+n=0 \: \: \: \: \: \: ⟶(3)$

$\sf \: ⇒a^2 −ma+n=0 \: \: \: \: \: \: \: ⟶(4)$

Subtracting (2) & (4) we get

$\sf \: −ap+am+q−n=0$

$\sf \: ⇒+a(m−p)=n−q$

$\pink{ \sf \: a = \frac{n - q}{m - p}}$

Hence, Proved.