Checkpoint: Solving with the RRT Solution

Determine all roots of the function. $f (x) = x^{3} + 3 x^{2} - 43 x + 15$

$RRT = \frac{\pm 1, \pm 3, \pm 5, \pm 15}{\pm 1} = \pm 1, \pm 3, \pm 5, \pm 15$

$\begin{array}{rcl} x^{2} + 8 x - 3 & = & 0 \\ a & = & 1, b = 8, c = - 3 \\ x & = & \frac{- 8 \pm \sqrt{64 - 4 (1) (- 3)}}{2 (1)} = \frac{- 8 \pm \sqrt{76}}{2} \\ = & \frac{- 8 \pm 2 \sqrt{19}}{2} = - 4 \pm \sqrt{19} \\ x & = & 5, - 4 \pm \sqrt{19} \end{array}$

Note

You can use technology to determine a starting point for synthetic division.

Q: What is another way to solve a quadratic equation besides factoring?

A: The quadratic formula or completing the square.