Question

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find all the zeros of the function. when there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. (enter your answers as a comma-separated list)
$f(x) = 9x^{3} - 42x^{2} + 34x - 40$
$x = square$

find all the zeros of the function. when there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. (enter your answers as a comma-separated list)
$g(x) = x^{5} - 8x^{4} + 30x^{3} - 68x^{2} + 88x - 48$
$x = square$
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Explanation:

First Function: $f(x) = 9x^3 - 42x^2 + 34x - 40$

Step1: List possible rational zeros

By Rational Root Theorem, possible zeros are $\pm1, \pm2, \pm4, \pm5, \pm8, \pm10, \pm20, \pm40, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{4}{3}, \pm\frac{5}{3}, \pm\frac{8}{3}, \pm\frac{10}{3}, \pm\frac{20}{3}, \pm\frac{40}{3}, \pm\frac{1}{9}, \pm\frac{2}{9}, \pm\frac{4}{9}, \pm\frac{5}{9}, \pm\frac{8}{9}, \pm\frac{10}{9}, \pm\frac{20}{9}, \pm\frac{40}{9}$

Step2: Test $x=4$ via synthetic division

$$\begin{array}{r|rrrr} 4 & 9 & -42 & 34 & -40 \\ & & 36 & -24 & 40 \\ \hline & 9 & -6 & 10 & 0 \end{array}$$

Since remainder is 0, $x=4$ is a zero.

Step3: Factor quadratic and find roots

The reduced polynomial is $9x^2 -6x +10$. Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$:
$$x=\frac{6\pm\sqrt{(-6)^2-4(9)(10)}}{2(9)}=\frac{6\pm\sqrt{36-360}}{18}=\frac{6\pm\sqrt{-324}}{18}=\frac{6\pm18i}{18}=\frac{1}{3}\pm i$$

Step1: List possible rational zeros

By Rational Root Theorem, possible zeros are $\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12, \pm16, \pm24, \pm48$

Step2: Test $x=2$ via synthetic division

$$\begin{array}{r|rrrrrr} 2 & 1 & -8 & 30 & -68 & 88 & -48 \\ & & 2 & -12 & 36 & -64 & 48 \\ \hline & 1 & -6 & 18 & -32 & 24 & 0 \end{array}$$

Remainder is 0, so $x=2$ is a zero.

Step3: Test $x=2$ again on reduced polynomial

$$\begin{array}{r|rrrrr} 2 & 1 & -6 & 18 & -32 & 24 \\ & & 2 & -8 & 20 & -24 \\ \hline & 1 & -4 & 10 & -12 & 0 \end{array}$$

Remainder is 0, so $x=2$ is a repeated zero.

Step4: Test $x=3$ on new reduced polynomial

$$\begin{array}{r|rrrr} 3 & 1 & -4 & 10 & -12 \\ & & 3 & -3 & 21 \\ \hline & 1 & -1 & 7 & 9 \end{array}$$

Remainder is not 0. Test $x=2$ again:

$$\begin{array}{r|rrrr} 2 & 1 & -4 & 10 & -12 \\ & & 2 & -4 & 12 \\ \hline & 1 & -2 & 6 & 0 \end{array}$$

Remainder is 0, so $x=2$ is a triple zero.

Step5: Factor quadratic and find roots

Reduced polynomial is $x^2-2x+6$. Use quadratic formula:
$$x=\frac{2\pm\sqrt{(-2)^2-4(1)(6)}}{2(1)}=\frac{2\pm\sqrt{4-24}}{2}=\frac{2\pm\sqrt{-20}}{2}=\frac{2\pm2i\sqrt{5}}{2}=1\pm i\sqrt{5}$$

Answer:

$x=4, \frac{1}{3}+i, \frac{1}{3}-i$

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Second Function: $g(x) = x^5 - 8x^4 + 30x^3 - 68x^2 + 88x - 48$