Question

which of the following is a complex zero of the polynomial function $p(x) = x^3 - x^2 - 14x - 30$? note: real numbers are a subset of the set of complex numbers. $2 + sqrt{2} cdot i$ $-2 - sqrt{2} cdot i$ $14$ $3$ question 19 2 pts find the average rate of change for the function $g(x) = \frac{1}{x + 3}$ on the interval $2, 2 + h$. $\frac{1}{8(8 + h)}$ $8(8 + h)$ $-\frac{1}{5(5 + h)}$ $-5(5 + h)$

Explanation:

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First Question (Polynomial Zero)

Step1: Test real zero candidate 3

$P(3) = 3^3 - 3^2 - 14(3) - 30 = 27 - 9 - 42 - 30 = -54
eq 0$

Step2: Test real zero candidate 14

$P(14) = 14^3 - 14^2 - 14(14) - 30 = 2744 - 196 - 196 - 30 = 2322
eq 0$

Step3: Factor polynomial via rational root test

Test $x=5$: $P(5)=125-25-70-30=0$. Factor as $(x-5)(x^2+4x+6)$

Step4: Solve quadratic for complex roots

Use quadratic formula: $x=\frac{-4\pm\sqrt{16-24}}{2}=\frac{-4\pm\sqrt{-8}}{2}=-2\pm\sqrt{2}i$

Step1: Recall average rate of change formula

$\text{Average rate} = \frac{g(2+h)-g(2)}{(2+h)-2}$

Step2: Compute $g(2+h)$ and $g(2)$

$g(2+h)=\frac{1}{(2+h)+3}=\frac{1}{5+h}$, $g(2)=\frac{1}{2+3}=\frac{1}{5}$

Step3: Substitute into the formula

$\frac{\frac{1}{5+h}-\frac{1}{5}}{h} = \frac{\frac{5-(5+h)}{5(5+h)}}{h}$

Step4: Simplify the expression

$\frac{\frac{-h}{5(5+h)}}{h} = -\frac{1}{5(5+h)}$

Answer:

$-2 - \sqrt{2} \cdot i$

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Second Question (Average Rate of Change)