Question

if both (x-2) and (x+1/2) are factors of px2+5x+r,show that p=r

Answer 1

X-2=0
X=2

X+1/2=0
X=-1/2


P(2)=0
P×(2)^2+5×2+r=0
4p+r=(-10)...........(1)


P(-1/2)=0
P×(-1/2)^2+5×(-1/2)+r=0
1/4p+r=-5/2............(2)

Solving 1 and 2 we get p=(-2) and r=(-2)

Hope this helps you

Answer 2

Seems like there is a mistake in your question. 2nd factor is (x - 1/2).

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