If the polynomials 2x3 + ax2 + 3x – 5 and x3 + x2 – 4x + a leave t

1.	If the polynomials 2x3 + ax2 + 3x – 5 and x3 + x2 – 4x + a leave the same remainder, when divided by x – 2, find the value of a.
Answer» Let f(x) = 2x³ + ax² + 3x – 5 g(x) = x³ + x² – 4x + a Given that f(x) and g(x) divided x – 2 give same remainder. i e., f(2) = g(2) By Remainder theorem. But f(2) = 2(2)³ + a(2) + 3(2)² – 5 = 2 x 8 + 4a + 6 – 5 = 17 +4a g(2) = 2³ + 2² – 4(2) + a. = 8 + 4 – 8 + a = 4 + a i.e., 4 + a = 17 + 4a ∴ a – 4a = 17 – 4 – 3a = 13 a = -13/3

If the polynomials 2x3 + ax2 + 3x – 5 and x3 + x2 – 4x + a leave the same remainder, when divided by x – 2, find the value of a.

Answer»

Let f(x) = 2x³ + ax² + 3x – 5

g(x) = x³ + x² – 4x + a

Given that f(x) and g(x) divided x – 2 give same remainder.

i e., f(2) = g(2)

By Remainder theorem.

But f(2) = 2(2)³ + a(2) + 3(2)² – 5

= 2 x 8 + 4a + 6 – 5

= 17 +4a

g(2) = 2³ + 2² – 4(2) + a.

= 8 + 4 – 8 + a = 4 + a

i.e., 4 + a = 17 + 4a

∴ a – 4a = 17 – 4 – 3a = 13

a = -13/3