If (x – 3) and both are factors of ax² + 5x + b, then show that a = b.

Let p(x) = ax² + 5x + b

As (x – 3) and are factors of p(x) hence each will divide p(x) leaving remainder as 0

Let us use the remainder theorem which states that if (x – a) divides a quadratic polynomial p(x) then p(a) = 0

⇒ p(3) = 0

Substituting x = 3

⇒ a(3)² + 5(3) + b = 0

⇒ 9a + 15 + b = 0

⇒ b = – 9a – 15 …(ii)

And also

Substituting

Multiplying throughout by 9

⇒ a + 15 + 9b = 0

⇒ –9b = a + 15 …(i)

Add (i) and (ii)

⇒ b – 9b = – 9a – 15 + a + 15

⇒ –8b = –8a

⇒ a = b

Hence proved