Question

If both x-2 and x-1/2 are factor of px^2+5x+r, show that p= r

Answer 1

x-2 and x-1/2 are factors so,

x=2 and x=1/2 are zeroes of polynomial by factor theoram.

$p {(2)}^{2} + 5(2) + r = 0 \\ 4p + 10+ r = 0 \\ \\ now \: by \: other \: zero \\ \\ p { (\frac{1}{2} )}^{2} + \frac{5}{2} + r = 0 \\ \frac{p}{4} + \frac{5}{2} + r = 0 \\ p + 10 + 4r = 0 \\ \\ 10 = - 4r - p \\ now \: put \: 10 \: in \: eq.1 \\ 4p + ( - 4r - p) + r = 0 \\ 3p = 3r \\ p = r$
hence proved.

since value of 10 is same in both case so we can easily replace 10.