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Chapter 2: Problem 2
Evaluate the given expression. Do not use a calculator. $$ 4^{3}-3^{4} $$
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Explain why the polynomial \(p\) defined by $$ p(x)=x^{6}+100 x^{2}+5 $$ has no real zeros.
Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (s(x))^{2} t(x) $$
Find all real numbers \(x\) such that $$ x^{4}+5 x^{2}-14=0 $$.
Suppose you start driving a car on a chilly fall day. As you drive, the heater in the car makes the temperature inside the car \(F(t)\) degrees Fahrenheit at time \(t\) minutes after you started driving, where $$ F(t)=40+\frac{30 t^{3}}{t^{3}+100} $$ (a) What was the temperature in the car when you started driving? (b) the car ten minutes after you started driving? (c) What will be the approximate temperature in the car after you have been driving for a long time?
Find the asymptotes of the graph of the given function \(\mathrm{r}\). $$ r(x)=\frac{9 x+5}{x^{2}-x-6} $$
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