Question

If n-k is a factor of the polynomials x^2+px +q and x^2+mx+n, prove that k = n+ n-k/m-p

Answer 1

Properties of common factors.

Answer 2

$\bold{k=\frac{n-q}{p-m}}$ is proven when n-k is factor of polynomials $\bold{x^{2}+p x+q\ and\ x^{2}+m x+n}$

Solution:

Let $f(x)=x^{2}+p x+q$ and $g(x)=x^{2}+m x+n$  

Given: f(x) and g(x) has factor (n-k).

If (x-a) is factor of f(x), then f (a) = 0.

So, f(k) = 0 ⇒ $k^{2}+p k+q=0$

And g(k) = 0⟹$k^{2}+m k+n=0$

Equating both the equations.

$\begin{array}{l}{k^{2}+p k+q=k^{2}+m k+n} \\ \\{p k-m k=n-q} \\ \\{k(p-m)=(n-q)} \\ \\{k=\frac{n-q}{p-m}}\end{array}$