Question

Find $$a$$ if the two polynomials $$ax^3 + 3x^2 - 9$$ and $$2x^3 + 4x + a$$ leave the same remainder when divided by $$(x + 3)$$.

Answer 1

Let $$f(x) = ax^3 + 3x^2 - 9$$ and $$g(x) = 2x^3 + 4x + a$$. 

Since the divisor is $$x + 3$$, the remainders are $$f(-3)$$ and $$g(-3)$$.

$$f(-3) = a(-3)^3 + 3(-3)^2 - 9 = -27a + 18,$$

$$g(-3) = 2(-3)^3 + 4(-3) + a = a - 66.$$

Since the remainders are equal,

$$-27a + 18 = a - 66$$

$$\Rightarrow 84 = 28a$$

$$\Rightarrow a = 3$$