Question

Without actual division, show that (x³ − 3x² − 13x + 15) is exactly divisible by (x² + 2x − 3).

Answer 1

$$\begin{gathered} x^3-3x^2-13x+15 \\ =x^3-5x^2+2x^2-10x-3x+15 \\ =x^2\left(x-5\right)+2x\left(x-5\right)-3\left(x-5\right) \\ =\left(x-5\right)\left(x^2+2x-3\right) \\ \mathrm{Here},x^2+2x-3\mathrm{is}\mathrm{a}\mathrm{factor}\mathrm{of}{x}^3-3x^2-13x+15. \\ \mathrm{Therefore},x^3-3x^2-13x+15\mathrm{is}\mathrm{exactly}\mathrm{divisible}\mathrm{by}{x}^2+2x-3. \end{gathered}$$