Question

Using the Remainder Theorem, find the remainders obtained when $$x^3 + (kx + 8)x + k$$ is divided by $$x + 1$$ and $$x - 2$$. Hence find $$k$$ if the sum of the two remainders is 1.

Answer 1

Let

$$f(x) = x^3 + (kx + 8)x + k = x^3 + kx^2 + 8x + k.$$

For the divisor $$x + 1$$, the remainder is

$$f(-1) = (-1)^3 + k(-1)^2 + 8(-1) + k = 2k - 9.$$

For the divisor $$x - 2$$, the remainder is

$$f(2) = 2^3 + k(2)^2 + 8(2) + k = 5k + 24.$$

Their sum is $$1$$, so

$$(2k - 9) + (5k + 24) = 1$$

$$\Rightarrow 7k + 15 = 1$$

$$\Rightarrow k = -2$$

Thus, the remainders are

$$2(-2) - 9 = -13$$

And

$$5(-2) + 24 = 14$$

Hence, $$k = -2$$ and the required remainders are $$-13$$ and $$14$$.