Question

b. determine the roots of $p(x) = x^3 - 5x^2 + 11x - 15$.

Explanation:

Step1: Test rational root candidates

By Rational Root Theorem, possible rational roots are $\pm1, \pm3, \pm5, \pm15$. Test $x=3$:
$$P(3)=3^3 - 5(3)^2 + 11(3) - 15 = 27 - 45 + 33 - 15 = 0$$
So $x=3$ is a root, and $(x-3)$ is a factor.

Step2: Polynomial division

Divide $P(x)=x^3-5x^2+11x-15$ by $(x-3)$:
$$\frac{x^3-5x^2+11x-15}{x-3}=x^2-2x+5$$

Step3: Solve quadratic equation

Solve $x^2-2x+5=0$ using quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=1, b=-2, c=5$:
$$x=\frac{2\pm\sqrt{(-2)^2-4(1)(5)}}{2(1)}=\frac{2\pm\sqrt{4-20}}{2}=\frac{2\pm\sqrt{-16}}{2}=\frac{2\pm4i}{2}=1\pm2i$$

Answer:

The roots are $3$, $1+2i$, and $1-2i$.