If two polynomials 2x3 + ax2 + 4x – 12 and x3 + x2 – 2x + a leave

1.	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» Let f(x) = 2x³ + ax² + 4x – 12 and g(x) = x³ + x² – 2x + a When f(x) is divided by x – 3, the remainder is f(3). Now f(3) = 2(3)³ + a(3)² + 4(3) – 12 = 54 + 9a + 12 – 12 f(3) = 9a + 54 … (1) When g(x) is divided by x – 3, the remainder is g(3). Now g(3) = 3³ + 3² – 2(3) + a = 27 + 9 – 6 + a g(3) = a + 30 … (2) Since, the remainder’s are same (1) = (2) Given that f(3) = g(3) That is 9a + 54 = a + 30 9a – a = 30 – 54 ⇒ 8a = -24 ∴ a = -3 Substituting a = -3 in f(3), we get f(3) = 9(-3) + 54 = -27 + 54 f(3) = 27 ∴ The remainder is 27

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»

Let f(x) = 2x³ + ax² + 4x – 12 and g(x) = x³ + x² – 2x + a

When f(x) is divided by x – 3, the remainder is f(3).

Now f(3) = 2(3)³ + a(3)² + 4(3) – 12 = 54 + 9a + 12 – 12

f(3) = 9a + 54 … (1)

When g(x) is divided by x – 3, the remainder is g(3).

Now g(3) = 3³ + 3² – 2(3) + a = 27 + 9 – 6 + a

g(3) = a + 30 … (2)

Since, the remainder’s are same (1) = (2)

Given that f(3) = g(3)

That is 9a + 54 = a + 30

9a – a = 30 – 54 ⇒ 8a = -24 ∴ a = -3

Substituting a = -3 in f(3), we get

f(3) = 9(-3) + 54 = -27 + 54

f(3) = 27

∴ The remainder is 27