Question

if both x-4 and x-1/4 are factors of ax^2+5x+b , show that a=b

Answer 1

let p(x)=ax²+5x+b,

since (x-4) is a factor of polynomial,
then
p(4)=0,

a×4²+5×4+b=0,

16a+20+b=0,.......eq(1),

since (x-1/4) is a factor of p(x), then

p(1/4) =0,

a×(1/4)²+5×1/4+b=0,

a/16 + 5/4 +b=0,

a+20 + 16b=0,.........eq(2),

on solving these two equations we will get

a=b