Beyond the Basics

43. There are three cube roots: $1, - \frac{1}{2} + \frac{\sqrt{3}}{2} i,$ $1, - \frac{1}{2} + \frac{\sqrt{3}}{2} i,$ and $- \frac{1}{2} - \frac{\sqrt{3}}{2} i$ $- \frac{1}{2} - \frac{\sqrt{3}}{2} i$
45. $P (x) = (x - 2) (x + i)$ $P (x) = (x - 2) (x + i)$
47. $P (x) = (x - 4) (x + 1) (x - i)$ $P (x) = (x - 4) (x + 1) (x - i)$
49. $f (x) = - 4 (x - 2) (x^{2} - 2 x + 5)$ $f (x) = - 4 (x - 2) (x^{2} - 2 x + 5)$

$y \to \infty as x \to - \infty, y \to - \infty as x \to \infty$ $y \to \infty as x \to - \infty, y \to - \infty as x \to \infty$
51. $f (x) = - 2 (x^{2} - 1) (x^{2} - 6 x + 10)$ $f (x) = - 2 (x^{2} - 1) (x^{2} - 6 x + 10)$

$y \to - \infty as x \to - \infty, y \to - \infty as x \to \infty$ $y \to - \infty as x \to - \infty, y \to - \infty as x \to \infty$

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