Question

What must be subtracted from the polynomial $$x^{3} + x^{2} - 2x + 1$$, so that the result is exactly divisible by $$(x - 3)$$?

Answer 1

Let the amount subtracted be $$k$$ and let $$f(x) = x^{3} + x^{2} - 2x + 1$$. 

For the resulting polynomial to be divisible by $$(x - 3)$$, its value at $$x = 3$$ must be zero.

$$f(3) = 3^3 + 3^2 - 2(3) + 1$$

$$= 27 + 9 - 6 + 1$$

$$= 31$$

Therefore, $$31 - k = 0$$, giving $$k = 31$$.

Indeed, $$x^{3} + x^{2} - 2x + 1 - 31 = (x - 3)(x^{2} + 4x + 10).$$