Question

If x-2 & x-1/2 both are the factors of polynomial nx^2-5x+m then prove m=n=2

Answer 1

let p(x)=nx²-5x+m,

since (x-2) is a factor of p(x), then
p(2)=0,

n(2)²-5(2)+m=0,

4n-10+m=0,

4n+m=10,.......eq(1),

again since (x-1/2) is also a factor of p(x), then
p(1/2)=0,

n(1/2)²-5(1/2) +m=0,

n/4 -5/2+m=0,

then
n-10+4m=0,

n+4m=10,........eq(2),


now from eq(1) and eq(2), we get

4n+m=10,
n+4m=10,

on solving these two equations we get
m=2 and n=2