Question

If the polynomials f(x) = ax³ + 4x² + 3x – 4 and g(x) = x³ – 4x + a leave the same remainder when divided by x – 3. Find the value of a. Also find the remainder.

Answer 1

Let f(x) = ax³ + 4x² + 3x – 4 and g(x) = x³ – 4x + a, 

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

Now f(3) = a(3)³ + 4(3)² + 3(3) – 4 = 27a + 36 + 9 – 4 

f(3) = 27a + 41 …(1) 

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

Now g(3) = 33 – 4(3) + a = 27 – 12 + a = 15 + a … (2) 

Since the remainders are same, (1) = (2) 

Given that, f(3) = g(3) 

That is 27a + 41 = 15 + a 

27a – a = 15 – 41 

26a = -26

\(a=\frac{-26}{26}=-1\)

Sustituting a = -1, in f(3), we get 

f(3) = 27(-1) + 41 = -27 + 41

f(3) = 14 

∴ The remainder is 14.