Remainder theorem

If the polynomial ax³ +4x² + 3x – 4 and x³ - 4x + a  leaves the same remainder when divided by (x – 3), then find the value of a?

Answer:

Let: F(x) = ax³ +4x² + 3x – 4

And G(x) = x³ - 4x + a  

Let the Remainder be y

Now, (F(x) – y) and (G(x) – y) are completely divisible by (x – 3)

Therefore, F(3) – y = 0 and G(3) – y = 0 and as per the remainder theorem.

On substitution we get F(3) – y ;

27a + 41 –y  = 0  ----------------------------------(1)

On substitution we get G(3) – y ;

15 + a – y = 0   -----------------------------------(2)

Equating (1) and (2), we get

27a + 41 = 15 + a

Or, 26a = 26

Or, a = 1