Question

If both x-2 and 2x-1 are factors of px²+5x+r show that p+r=0

Answer 1

Step-by-step explanation:

Given : x - 2 and 2x - 1 are the factors of the polynomial given below :

f(x) = px² + 5x + r

If (x - 2) is a factor, then

x - 2 = 0

x = 2

So, f(2) = 0

f(2) = p(2)² + 5(2) + r

0 = 4p + 10 + r

4p + r = - 10 ...(i)

If (2x - 1) is a factor, then

2x - 1 = 0

x = 1/2

So, f(1/2) = 0

f(1/2) = p(1/2)² + 5(1/2) + r

0 = a/p + 5/2 + r

0 = (p + 10 + 4r)/4

p + 4r = - 10 ...(ii)

Multiplying (ii) by 4, we get

4(p + 4r) = 4(- 10)

4p + 16r = - 40 ...(iii)

Subtracting (i) and (iii), we get

→ - 15r = 30

→ r = - 2

Put this value in (ii), we get

→ p + 4(- 2) = - 10

→ p + (-8) = - 10

→ p = - 10 + 8

→ p = - 2

Now,

p - r = - 2 - (- 2)

= - 2 + 2

= 0

Hence, proved !!