Question

If both x - 2 and x-1/2 are factors of px2

+ 5x + r, show that p = r.​

Answer 1

As both are the factors of the given polynomials, both must be zero for x = 2 and 1/2.   Using factor theorem:

 If x - 2 is factor: f(2) = 0

⇒ p(2)² + 5(2) + r = 0

⇒ 4p + 10 + r = 0          ...(1)

 If x - 1/2 is factor: f(1/2) = 0

⇒ p(1/2)² + 5(1/2) + r = 0

⇒ p/4 + 5/2 + r = 0

⇒ p + 10 + 4r = 0         ...(2)

   Subtract (1) from (2), we get:

⇒ 3p - 3r = 0

⇒ 3p = 3r

⇒ p = r      proved