Question

If the polynomials $$(2x^3 + ax^2 + 3x - 5)$$ and $$(x^3 + x^2 - 2x + a)$$ leave the same remainder when divided by $$(x - 2)$$, find the value of $$a$$. Also, find the remainder in each case.

Answer 1

Let $$P(x) = 2x^3 + ax^2 + 3x - 5$$ and $$Q(x) = x^3 + x^2 - 2x + a$$.

By the Remainder Theorem,

$$P(2) = 2(2)^3 + a(2)^2 + 3(2) - 5 = 4a + 17,$$

$$Q(2) = (2)^3 + (2)^2 - 2(2) + a = 8 + a.$$

Since the remainders are equal,

$$4a + 17 = 8 + a$$

$$\Rightarrow 3a = -9$$

$$\Rightarrow a = -3$$

The remainder in each case is

$$8 + a = 8 - 3$$

$$= 5$$