Question

if x+a is a factor of polynomial x^2 + lx+ m and x^2 +nx +k then prove that a=m-k / l-k

Answer 1

$x + a = 0 \\ x = - a \\ p(x) = {x}^{2} + lx + m \\ p( - a) = ( { - a}^{2} ) + l( - a) + m \\ {a}^{2} - la + m = 0.....1 \\ and \\ p(x) = {x}^{2} + nx + k \\ p( - a) = { - a}^{2} + n( - a) + k \\ {a}^{2} - na + k = 0.....2 \\ from \: 1 \: and \: 2 \\ {a}^{2} - la + m = {a}^{2} - na + k \\ - la + na = - m + k \\ - a(l - n) = - (m - k) \\ a = \frac{m - k}{l - n}$