Chapter 4

Solve the polynomial inequality (a) symbolically and (b) graphically. $$ 4 x^{4}-5 x^{2}-9 \geq 0 $$

Transformations Use transformations of the graph of either \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^{2}}\) to sketch a graph of \(y=g(x)\) by hand. Show all asymptotes. Write \(g(x)\) in terms of either \(f(x)\) or \(h(x)\) $$ g(x)=\frac{1}{x+2} $$

Use the remainder theorem to find the remainder when \(f(x)\) is divided by the given \(x-k\) $$f(x)=5 x^{2}-3 x+1 \quad\quad\quad x-1$$

Electricity \(\quad\) Complex numbers are used in the study of electrical circuits. Impedance \(Z\) (or the opposition to the flow of electricity. voltage \(V\) and current \(I\) can all be represented by complex numbers. They are related by the equation \(Z=\frac{V}{I} .\) Find the value of the missing variable. $$ Z=22-5 i \quad V=27+17 i $$

Solve the equation. Check your answers. $$ \sqrt[3]{x+1}=\sqrt[3]{2 x-1} $$

Solve the polynomial inequality. $$ 7 x^{4}>14 x^{2} $$

Transformations Use transformations of the graph of either \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^{2}}\) to sketch a graph of \(y=g(x)\) by hand. Show all asymptotes. Write \(g(x)\) in terms of either \(f(x)\) or \(h(x)\) $$ g(x)=\frac{1}{x}+2 $$

Use the remainder theorem to find the remainder when \(f(x)\) is divided by the given \(x-k\) $$f(x)=-4 x^{2}+6 x-7 \quad\quad\quad x+4$$

Solve the equation. Check your answers. $$ \sqrt[3]{2 x^{2}+1}=\sqrt[3]{1-x} $$

Solve the polynomial inequality. $$ 3 x^{4}-4 x^{2}<7 $$

Transformations Use transformations of the graph of either \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^{2}}\) to sketch a graph of \(y=g(x)\) by hand. Show all asymptotes. Write \(g(x)\) in terms of either \(f(x)\) or \(h(x)\) $$ g(x)=1-\frac{2}{x} $$

Use the remainder theorem to find the remainder when \(f(x)\) is divided by the given \(x-k\) $$f(x)=4 x^{3}-x^{2}+4 x+2 \quad x+2$$

Solve the equation. Check your answers. $$ \sqrt[4]{x-2}+4=20 $$

Solve the polynomial inequality. $$ (x-1)(x-2)(x+2) \geq 0 $$

Transformations Use transformations of the graph of either \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^{2}}\) to sketch a graph of \(y=g(x)\) by hand. Show all asymptotes. Write \(g(x)\) in terms of either \(f(x)\) or \(h(x)\) $$ g(x)=\frac{1}{x+1}-2 $$

Use the remainder theorem to find the remainder when \(f(x)\) is divided by the given \(x-k\) $$f(x)=-x^{4}+4 x^{3}-x+3 \quad x-3$$

Give an example of a polynomial function that has only imaginary zeros and a polynomial function that has only real zeros. Explain how to determine graphically if a function has only imaginary zeros.

Solve the equation. Check your answers. $$ \sqrt[4]{2 x+3}=\sqrt{x+1} $$

Solve the polynomial inequality. $$ -(x+1)^{2}(x-2) \geq 0 $$

Transformations Use transformations of the graph of either \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^{2}}\) to sketch a graph of \(y=g(x)\) by hand. Show all asymptotes. Write \(g(x)\) in terms of either \(f(x)\) or \(h(x)\) $$ g(x)=\frac{1}{x-2}+1 $$

Evaluate \(f(x)\) at the given \(x\) Approximate each result to the nearest hundredth. $$ f(x)=x^{3 / 2}-x^{1 / 2}, \quad x=50 $$

Solve the polynomial inequality. $$ 2 x^{4}+2 x^{3} \leq 12 x^{2} $$

Transformations Use transformations of the graph of either \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^{2}}\) to sketch a graph of \(y=g(x)\) by hand. Show all asymptotes. Write \(g(x)\) in terms of either \(f(x)\) or \(h(x)\) $$ g(x)=-\frac{2}{(x-1)^{2}} $$

Evaluate \(f(x)\) at the given \(x\) Approximate each result to the nearest hundredth. $$ f(x)=x^{5 / 4}-x^{-3 / 4}, \quad x=7 $$

Solve the polynomial inequality. $$ x^{3}+6 x^{2}+9 x>0 $$

Transformations Use transformations of the graph of either \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^{2}}\) to sketch a graph of \(y=g(x)\) by hand. Show all asymptotes. Write \(g(x)\) in terms of either \(f(x)\) or \(h(x)\) $$ g(x)=\frac{1}{x^{2}}-1 $$

Solve the polynomial inequality graphically. $$ x^{3}-7 x^{2}+14 x \leq 8 $$

Transformations Use transformations of the graph of either \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^{2}}\) to sketch a graph of \(y=g(x)\) by hand. Show all asymptotes. Write \(g(x)\) in terms of either \(f(x)\) or \(h(x)\) $$ g(x)=\frac{1}{(x+1)^{2}}-2 $$

When can you use synthetic division to divide two polynomials? Give one example where synthetic division can be used and one example where it cannot be used.

Solve the polynomial inequality graphically. $$ 2 x^{3}+3 x^{2}-3 x<2 $$

Transformations Use transformations of the graph of either \(f(x)=\frac{1}{x}\) or \(h(x)=\frac{1}{x^{2}}\) to sketch a graph of \(y=g(x)\) by hand. Show all asymptotes. Write \(g(x)\) in terms of either \(f(x)\) or \(h(x)\) $$ g(x)=1-\frac{1}{(x-2)^{2}} $$

Solve the polynomial inequality graphically. $$ 3 x^{4}-7 x^{3}-2 x^{2}+8 x>0 $$

Complete the following. (a) Find the domain of \(f\) (b) Graph \(f\) in an appropriate viewing rectangle. (c) Find any horizontal or vertical asymptotes. (d) Sketch a graph of \(f\) that includes any asymptotes. $$ f(x)=\frac{x+3}{x-2} $$

Use translations to graph \(f .\) $$ f(x)=\sqrt[3]{x-1} $$

Solve the polynomial inequality graphically. $$ x^{4}-5 x^{3} \leq 5 x^{2}+45 x+36 $$

Complete the following. (a) Find the domain of \(f\) (b) Graph \(f\) in an appropriate viewing rectangle. (c) Find any horizontal or vertical asymptotes. (d) Sketch a graph of \(f\) that includes any asymptotes. $$ f(x)=\frac{6-2 x}{x+3} $$

Use translations to graph \(f .\) $$ f(x)=x^{2 / 3}-1 $$

Solve the rational inequality (a) symbolically and (b) graphically. $$ \frac{1}{x}<0 $$

Complete the following. (a) Find the domain of \(f\) (b) Graph \(f\) in an appropriate viewing rectangle. (c) Find any horizontal or vertical asymptotes. (d) Sketch a graph of \(f\) that includes any asymptotes. $$ f(x)=\frac{4 x+1}{x^{2}-4} $$

Use translations to graph \(f .\) $$ f(x)=\sqrt{x-1} $$

Solve the rational inequality (a) symbolically and (b) graphically. $$ \frac{1}{x^{2}}>0 $$

Complete the following. (a) Find the domain of \(f\) (b) Graph \(f\) in an appropriate viewing rectangle. (c) Find any horizontal or vertical asymptotes. (d) Sketch a graph of \(f\) that includes any asymptotes. $$ f(x)=\frac{0.5 x^{2}+1}{x^{2}-9} $$

Use translations to graph \(f .\) $$ f(x)=\sqrt{x+2}-1 $$

Solve the rational inequality (a) symbolically and (b) graphically. $$ \frac{4}{x+3} \geq 0 $$

Complete the following. (a) Find the domain of \(f\) (b) Graph \(f\) in an appropriate viewing rectangle. (c) Find any horizontal or vertical asymptotes. (d) Sketch a graph of \(f\) that includes any asymptotes. $$ f(x)=\frac{4}{1-0.25 x^{2}} $$

Solve the rational inequality (a) symbolically and (b) graphically. $$ \frac{x-1}{x+1}<0 $$

Complete the following. (a) Find the domain of \(f\) (b) Graph \(f\) in an appropriate viewing rectangle. (c) Find any horizontal or vertical asymptotes. (d) Sketch a graph of \(f\) that includes any asymptotes. $$ f(x)=\frac{x^{2}}{1+0.25 x^{2}} $$

Solve the equation. Check your answers. $$ x^{3}=8 $$

Solve the rational inequality (a) symbolically and (b) graphically. $$ \frac{5}{x^{2}-4}<0 $$

Complete the following. (a) Find the domain of \(f\) (b) Graph \(f\) in an appropriate viewing rectangle. (c) Find any horizontal or vertical asymptotes. (d) Sketch a graph of \(f\) that includes any asymptotes. $$ f(x)=\frac{x^{2}-4}{x-2} $$