Question

If two polynomials 2x³ + ax² + 4x – 12 and x³ + x² – 2x + a leave the same remainder when divided by (x – 3), find the value of a and also find the remainder.

Answer 1

p(x₁) = 2x³ + ax² + 4x – 12

When it is divided by x – 3,

p(3) = 2(3)³ + a(3)² + 4(3) – 12

= 54 + 9a + 12 – 12

= 54 + 9a  ...(R₁)

p(x₂) = x³ + x² – 2x + a

When it is divided by x – 3,

p(3) = 3³ + 3² – 2(3) + a

= 27 + 9 – 6 + a

= 30 + a  ...(R₂)

The given remainders are same (R₁ = R₂)

∴ 54 + 9a = 30 + a

9a – a = 30 – 54

8a = – 24

∴ a = `(– 24)/8`

= – 3

Consider R₂,

Remainder = 30 – 3

= 27