Question

if x+a is a common factor of f(x)=x^(2) +px+q and g(x)=x^(2) +lx+m . Show that: a=(q-m)/(p-l)

Answer 1

$\textbf{Factor theorem:}$

$\text{(x-a) is a factor of f(x) iff f(a) =0}$

$\textbf{Given:}$

$\text{ x+a is a common factor of}$

$\text{ f(x)=x^2+px+q and g(x)=x^2+lx+m }$

$\text{Then, by factor theorem,}$

$\text{ f(-a)=0 and g(-a)=0 }$

$(-a)^2+p(-a)+q=0\;\text{and}\;(-a)^2+l(-a)+m=0$

$a^2-pa+q=0......(1)\;\text{and}\;a^2-la+m=0.....(2)$

$\text{subtracting (2) from (1), we get}$

$-pa+la+q-m=0$

$-pa+la=-(q-m)$

$-a(p-l)=-(q-m)$

$a(p-l)=q-m$

$\implies\boxed{\bf\,a=\frac{q-m}{p-l}}$

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