Question

If x-2and2x-1 are factors of ax^2+5x+b show that a-b=0

Answer 1

Step-by-step explanation:

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

f(x) = ax² + 5x + b

If (x - 2) is a factor, then

x - 2 = 0

x = 2

So, f(2) = 0

f(2) = a(2)² + 5(2) + b

0 = 4a + 10 + b

4a + b = - 10 ...(i)

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

2x - 1 = 0

x = 1/2

So, f(1/2) = 0

f(1/2) = a(1/2)² + 5(1/2) + b

0 = a/4 + 5/2 + b

0 = (a + 10 + 4b)/4

a + 4b = - 10 ...(ii)

Multiplying (ii) by 4, we get

4(a + 4b) = 4(- 10)

4a + 16b = - 40 ...(iii)

Subtracting (i) and (iii), we get

→ - 15b = 30

→ b = - 2

Put this value in (ii), we get

→ a + 4(- 2) = - 10

→ a + (-8) = - 10

→ a = - 10 + 8

→ a = - 2

Now,

a - b = - 2 - (- 2)

= - 2 + 2

= 0

Hence, proved !!