Question

The polynomials $$f(x) = (ax^3 + 3x^2 - 3)$$ and $$g(x) = (2x^3 - 5x + a)$$ when divided by $$(x - 4)$$ leave the same remainder in each case. Find the value of $$a$$.

Answer 1

By the Remainder Theorem,

$$f(4) = a(4)^3 + 3(4)^2 - 3 = 64a + 45,$$

$$g(4) = 2(4)^3 - 5(4) + a = 108 + a.$$

Since the remainders are equal,

$$64a + 45 = 108 + a$$

$$\Rightarrow 63a = 63$$

$$\Rightarrow a = 1$$