Question

1. 5 is a solution of the following polynomial. find all other solutions using synthetic division.

$f(x) = x^3 - 15x^2 + 44x + 30$

Explanation:

Step1: Set up synthetic division

Use root $x=5$, coefficients $1, -15, 44, 30$:

$$\begin{array}{r|rrrr} 5 & 1 & -15 & 44 & 30 \\ & & 5 & -50 & -30 \\ \hline & 1 & -10 & -6 & 0 \\ \end{array}$$

Step2: Get quadratic factor

From the result, the quadratic is $x^2 -10x -6$.

Step3: Solve quadratic equation

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=1, b=-10, c=-6$:

$$ x=\frac{10\pm\sqrt{(-10)^2-4(1)(-6)}}{2(1)}=\frac{10\pm\sqrt{100+24}}{2}=\frac{10\pm\sqrt{124}}{2}=\frac{10\pm2\sqrt{31}}{2}=5\pm\sqrt{31} $$

Answer:

The other solutions are $5+\sqrt{31}$ and $5-\sqrt{31}$; all solutions are $5$, $5+\sqrt{31}$, and $5-\sqrt{31}$.