Question

If ( x - 2 ) and ( 2x - 1 ) are factors of ax^2 + 5x + b , then show that a - b = 0.​

Answer 1

If (x-2) and (2x-1) are factors, then polynomial will be

(x-2)(2x-1)=0

⇒2x² -x -4x +2=0

⇒2x²-5x +2 =0

Hence, a = 2 and b = 2

so, a-b = 0

Answer 2

Given:-

• P(x) = ax² + 5x + b

• f(x) = (x - 2) , (2x - 1)

To Show:-

• a - b = 0

Now,

→ f(x) = x - 2 = 0

→ x = 2

Now Putting the value of f(x) in p(x).

→ P(x) = ax² + 5x + b

→ P(2) = a(2)² + 5(2) + b

→ 4a + 10 + b = 0

→ 4a + b = -10...........eq.1

Again

→ f(x) = 2x - 1

→ f(x) = 2x - 1 = 0

→ 2x = 1

→ x = 1/2

Now Putting the value of f(x) in p(x).

Therefore

→ P(x) = ax^2 + 5x + b

→ P(½) = a (½)² + 5(½) + b

→ a/4 + 5/2 + b = 0

→ (a + 10 + 4b)/4 = 0

→ a + 10 + 4b = 0 × 4

→ a + 10 + 4b = 0

→ a + 4b = -10.......eq2

Now, Subtracting the eq.1 and eq.2

→ 4a + b -( a + 4b) = -10-(-10)

→ 4a + b - a - 4b = -10 + 10

→ 3a - 3b = 0

→ 3 ( a - b ) = 0

→ a - b = 0

Hence, Proved ✔️