Chapter 4

Divide the first polynomial by the second. State the quotient and remainder. $$2 x^{4}-7 x^{3}-5 x^{2}-19 x+17 \quad\quad\quad x+1$$

Let \(a_{n}\) be the leading coefficient. (a) Find the complete factored form of a polynomial with real coefficients \(f(x)\) that satisfies the conditions. (b) Express \(f(x)\) in expanded form. Degree \(3 ; a_{n}=-1 ; \quad\) zeros \(-1,2 i,\) and \(-2 i\)

Evaluate the expression by hand. $$ 27^{5 / 6} \cdot 27^{-1 / 6} $$

Find all real solutions. Check your results. $$ \frac{1}{x+2}+\frac{1}{x}=1 $$

Determine whether \(f\) is a rational function and state its domain. $$ f(x)=4-\frac{3}{x+1} $$

Divide the first polynomial by the second. State the quotient and remainder. $$x^{4}-x^{3}-4 x+1 \quad\quad\quad x-2$$

Let \(a_{n}\) be the leading coefficient. (a) Find the complete factored form of a polynomial with real coefficients \(f(x)\) that satisfies the conditions. (b) Express \(f(x)\) in expanded form. $$ \text { Degree } 4 ; a_{n}=3 ; \quad \text { zeros }-2,4, i, \text { and }-i $$

Evaluate the expression by hand. $$ 16^{2 / 3} \cdot 16^{-1 / 6} $$

Find all real solutions. Check your results. $$ \frac{2 x}{x-1}=5+\frac{2}{x-1} $$

Determine whether \(f\) is a rational function and state its domain. $$ f(x)=5 x^{3}-4 x $$

Divide the first polynomial by the second. State the quotient and remainder. $$3 x^{3}-7 x+10 \quad\quad\quad x-1$$

Let \(a_{n}\) be the leading coefficient. (a) Find the complete factored form of a polynomial with real coefficients \(f(x)\) that satisfies the conditions. (b) Express \(f(x)\) in expanded form. $$ \text { Degree } 4 ; a_{n}=10 ; \quad \text { zeros } 1,-1,3 i, \text { and }-3 i $$

Evaluate the expression by hand. $$ (-27)^{-5 / 3} $$

Find all real solutions. Check your results. $$ \frac{1}{x}-\frac{2}{x^{2}}=5 $$

Divide the first polynomial by the second. State the quotient and remainder. $$x^{4}-16 x^{2}+1 \quad\quad\quad x+4$$

Let \(a_{n}\) be the leading coefficient. (a) Find the complete factored form of a polynomial with real coefficients \(f(x)\) that satisfies the conditions. (b) Express \(f(x)\) in expanded form. Degree \(2 ; a_{n}=-5 ; \quad\) zeros \(1+i\) and \(1-i\)

Evaluate the expression by hand. $$ (-32)^{-3 / 5} $$

Find all real solutions. Check your results. $$ \frac{1}{x^{2}-2}=\frac{1}{x} $$

Divide Check your answer. $$\frac{x^{4}-3 x^{3}-x+3}{x-3}$$

Let \(a_{n}\) be the leading coefficient. (a) Find the complete factored form of a polynomial with real coefficients \(f(x)\) that satisfies the conditions. (b) Express \(f(x)\) in expanded form. Degree \(4 ; a_{n}=\frac{1}{2}\) zeros \(-i\) and \(2 i\)

Evaluate the expression by hand. $$ \left(0.5^{-2}\right)^{2} $$

Find all real solutions. Check your results. $$ \frac{x^{3}-4 x}{x^{2}+1}=0 $$

Divide Check your answer. $$\frac{x^{3}-2 x^{2}-x+3}{x+1}$$

Evaluate the expression by hand. $$ \left(2^{-2}\right)^{-3 / 2} $$

Find all real solutions. Check your results. $$ \frac{1}{x+2}+\frac{1}{x+3}=\frac{2}{x^{2}+5 x+6} $$

Divide Check your answer. $$\frac{4 x^{3}-x^{2}-5 x+6}{x-1}$$

Let \(a_{n}\) be the leading coefficient. (a) Find the complete factored form of a polynomial with real coefficients \(f(x)\) that satisfies the conditions. (b) Express \(f(x)\) in expanded form. Degree \(3 ; a_{n}=-2 ; \quad\) zeros \(1-i\) and 3

Evaluate the expression by hand. $$ \left(\frac{2}{3}\right)^{-2} $$

Find all real solutions. Check your results. $$ \frac{35}{x^{2}}=\frac{4}{x}+15 $$

Divide Check your answer. $$\frac{x^{4}+3 x^{3}-4 x+1}{x+2}$$

Let \(a_{n}\) be the leading coefficient. (a) Find the complete factored form of a polynomial with real coefficients \(f(x)\) that satisfies the conditions. (b) Express \(f(x)\) in expanded form. Degree \(4 ; a_{n}=7\) zeros \(2 i\) and \(3 i\)

Evaluate the expression by hand. $$ \left(8^{-1 / 3}+27^{-1 / 3}\right)^{2} $$

Find all real solutions. Check your results. $$ 6-\frac{35}{x}+\frac{36}{x^{2}}=0 $$

Divide Check your answer. $$\frac{x^{3}+1}{x+1}$$

Use positive exponents to rewrite. $$ \sqrt{2 x} $$

Find all real solutions. Check your results. $$ \frac{x+5}{x+2}=\frac{x-4}{x-10} $$

Divide Check your answer. $$\frac{x^{5}+3 x^{4}-x-3}{x+3}$$

Use positive exponents to rewrite. $$ \sqrt{x+1} $$

Find all real solutions. Check your results. $$ \frac{x-1}{x+1}=\frac{x+3}{x-4} $$

Divide Check your answer. $$\frac{6 x^{3}+5 x^{2}-8 x+4}{2 x-1}$$

Use positive exponents to rewrite. $$ \sqrt[3]{z^{5}} $$

Find all real solutions. Check your results. $$ \frac{1}{x-2}-\frac{2}{x-3}=\frac{-1}{x^{2}-5 x+6} $$

Find any horizontal or vertical asymptotes. $$ f(x)=\frac{4 x+1}{2 x-6} $$

Divide Check your answer. $$\frac{12 x^{3}-14 x^{2}+7 x-7}{3 x-2}$$

Use positive exponents to rewrite. $$ \sqrt[5]{x^{2}} $$

Find all real solutions. Check your results. $$ \frac{1}{x-1}+\frac{3}{x+1}=\frac{4}{x^{2}-1} $$

Find any horizontal or vertical asymptotes. $$ f(x)=\frac{x+6}{5-2 x} $$

Divide the expression. $$\frac{3 x^{4}-7 x^{3}+6 x-16}{3 x-7}$$

Complete the following. (a) Find all zeros of \(f(x)\) (b) Write the complete factored form of \(f(x)\) $$ f(x)=x^{2}+25 $$

Use positive exponents to rewrite. $$ (\sqrt[4]{y})^{-3} $$