Question

If x-2 and x-1/2 are factors of p$x^2$ + 5x + r, show that p=r

Answer 1

Let f(x) = px² + 5x + r
(x - 2) is a factor of f(x), so f(2) = 0
=> 4p + 10 + r = 0 ... (1)

Also, (x - 1/2) is a factor of f(x) so f(1/2) = 0
=> p/4 + 5/2 + r = 0
=> p + 10 + 4r = 0 ... (2)
Subtracting (2) from (1), we get,
3p - 3r = 0
or, p = r

Therefore, it is proved that p = r.