Chapter 4

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{4}+4 x^{2} $$

A rational function is given. (a) Complete each table for the function. (b) Describe the behavior of the function nea its vertical asymptote, based on Tables 1 and \(2 .\) (c) Determine the horizontal asymptote, based on Tables 3 and \(4 .\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 1.5 & {} \\ {1.9} & {} \\\ {1.99} & {} & {} \\ {1.999} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 2.5 & {} \\ {2.1} & {} & {} \\\ {2.01} & {} \\ {2.001} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 10 & {} \\ {50} & {} \\\ {100} & {} \\ {1000} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline-10 & {} \\ {-50} & {} \\ {-100} & {} \\ {-1000} & {} \\\ \hline\end{array}\) \(r(x)=\frac{x}{x-2}\)

List all possible rational zeros given by the Rational Zeros Theorem (but don’t check to see which actually are zeros). $$ P(x)=x^{3}-4 x^{2}+3 $$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\). \(P(x)=3 x^{2}+5 x-4, \quad D(x)=x+3\)

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{5}+9 x^{3} $$

A rational function is given. (a) Complete each table for the function. (b) Describe the behavior of the function nea its vertical asymptote, based on Tables 1 and \(2 .\) (c) Determine the horizontal asymptote, based on Tables 3 and \(4 .\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 1.5 & {} \\ {1.9} & {} \\\ {1.99} & {} & {} \\ {1.999} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 2.5 & {} \\ {2.1} & {} & {} \\\ {2.01} & {} \\ {2.001} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 10 & {} \\ {50} & {} \\\ {100} & {} \\ {1000} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline-10 & {} \\ {-50} & {} \\ {-100} & {} \\ {-1000} & {} \\\ \hline\end{array}\) \(r(x)=\frac{4 x+1}{x-2}\)

List all possible rational zeros given by the Rational Zeros Theorem (but don’t check to see which actually are zeros). $$ Q(x)=x^{4}-3 x^{3}-6 x+8 $$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\). \(P(x)=x^{3}+4 x^{2}-6 x+1, \quad D(x)=x-1\)

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{3}-2 x^{2}+2 x $$

A rational function is given. (a) Complete each table for the function. (b) Describe the behavior of the function nea its vertical asymptote, based on Tables 1 and \(2 .\) (c) Determine the horizontal asymptote, based on Tables 3 and \(4 .\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 1.5 & {} \\ {1.9} & {} \\\ {1.99} & {} & {} \\ {1.999} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 2.5 & {} \\ {2.1} & {} & {} \\\ {2.01} & {} \\ {2.001} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 10 & {} \\ {50} & {} \\\ {100} & {} \\ {1000} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline-10 & {} \\ {-50} & {} \\ {-100} & {} \\ {-1000} & {} \\\ \hline\end{array}\) \(r(x)=\frac{3 x-10}{(x-2)^{2}}\)

List all possible rational zeros given by the Rational Zeros Theorem (but don’t check to see which actually are zeros). $$ R(x)=2 x^{5}+3 x^{3}+4 x^{2}-8 $$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\). \(P(x)=2 x^{3}-3 x^{2}-2 x, \quad D(x)=2 x-3\)

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{3}+x^{2}+x $$

A rational function is given. (a) Complete each table for the function. (b) Describe the behavior of the function nea its vertical asymptote, based on Tables 1 and \(2 .\) (c) Determine the horizontal asymptote, based on Tables 3 and \(4 .\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 1.5 & {} \\ {1.9} & {} \\\ {1.99} & {} & {} \\ {1.999} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 2.5 & {} \\ {2.1} & {} & {} \\\ {2.01} & {} \\ {2.001} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline 10 & {} \\ {50} & {} \\\ {100} & {} \\ {1000} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline x & {r(x)} \\ \hline-10 & {} \\ {-50} & {} \\ {-100} & {} \\ {-1000} & {} \\\ \hline\end{array}\) \(r(x)=\frac{3 x^{2}+1}{(x-2)^{2}}\)

List all possible rational zeros given by the Rational Zeros Theorem (but don’t check to see which actually are zeros). $$ S(x)=6 x^{4}-x^{2}+2 x+12 $$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\). \(P(x)=4 x^{3}+7 x+9, \quad D(x)=2 x+1\)

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{4}+2 x^{2}+1 $$

Find the \(x\) -and \(y\) -intercepts of the rational function. \(r(x)=\frac{x-1}{x+4}\)

List all possible rational zeros given by the Rational Zeros Theorem (but don’t check to see which actually are zeros). $$ T(x)=4 x^{4}-2 x^{2}-7 $$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\). \(P(x)=x^{4}-x^{3}+4 x+2, \quad D(x)=x^{2}+3\)

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{4}-x^{2}-2 $$

Find the \(x\) -and \(y\) -intercepts of the rational function. \(s(x)=\frac{3 x}{x-5}\)

List all possible rational zeros given by the Rational Zeros Theorem (but don’t check to see which actually are zeros). $$ U(x)=12 x^{5}+6 x^{3}-2 x-8 $$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\). \(P(x)=2 x^{5}+4 x^{4}-4 x^{3}-x-3, \quad D(x)=x^{2}-2\)

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{4}-16 $$

Find the \(x\) -and \(y\) -intercepts of the rational function. \(t(x)=\frac{x^{2}-x-2}{x-6}\)

A polynomial function \(P\) and its graph are given. (a) List all possible rational zeros of \(P\) given by the Rational Zeros Theorem. (b) From the graph, determine which of the possible rational zeros actually turn out to be zeros. $$ P(x)=5 x^{3}-x^{2}-5 x+1 $$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ \(P(x)=x^{2}+4 x-8, \quad D(x)=x+3\)

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{4}+6 x^{2}+9 $$

Find the \(x\) -and \(y\) -intercepts of the rational function. \(r(x)=\frac{2}{x^{2}+3 x-4}\)

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ \(P(x)=x^{3}+6 x+5, \quad D(x)=x-4\)

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{3}+8 $$

Find the \(x\) -and \(y\) -intercepts of the rational function. \(r(x)=\frac{x^{2}-9}{x^{2}}\)

A polynomial function \(P\) and its graph are given. (a) List all possible rational zeros of \(P\) given by the Rational Zeros Theorem. (b) From the graph, determine which of the possible rational zeros actually turn out to be zeros. $$ P(x)=2 x^{4}-9 x^{3}+9 x^{2}+x-3 $$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ \(P(x)=4 x^{2}-3 x-7, \quad D(x)=2 x-1\)

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{3}-8 $$

Find the \(x\) -and \(y\) -intercepts of the rational function. \(r(x)=\frac{x^{3}+8}{x^{2}+4}\)

A polynomial function \(P\) and its graph are given. (a) List all possible rational zeros of \(P\) given by the Rational Zeros Theorem. (b) From the graph, determine which of the possible rational zeros actually turn out to be zeros. $$ P(x)=4 x^{4}-x^{3}-4 x+1 $$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ \(P(x)=6 x^{3}+x^{2}-12 x+5, \quad D(x)=3 x-4\)

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{6}-1 $$

Find all rational zeros of the polynomial. $$ P(x)=x^{3}+3 x^{2}-4 $$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ \(P(x)=2 x^{4}-x^{3}+9 x^{2}, \quad D(x)=x^{2}+4\)

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=(x-1)(x+2) $$

\(1-12\) . A polynomial \(P\) is given. (a) Find all zeros of \(P\) , real and complex. (b) Factor \(P\) completely. $$ P(x)=x^{6}-7 x^{3}-8 $$

Find all rational zeros of the polynomial. $$ P(x)=x^{3}-7 x^{2}+14 x-8 $$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ \(P(x)=x^{5}+x^{4}-2 x^{3}+x+1, \quad D(x)=x^{2}+x-1\)

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=(x-1)(x+1)(x-2) $$

13- 30 . Factor the polynomial completely and find all its zeros. State the multiplicity of each zero. \(P(x)=x^{2}+25\)

Find the quotient and remainder using long division. \(\frac{x^{2}-6 x-8}{x-4}\)

Find all rational zeros of the polynomial. $$ P(x)=x^{3}-3 x-2 $$