Chapter 5

For the following exercises, list all possible rational zeros for the functions. $$ f(x)=2 x^{3}+3 x^{2}-8 x+5 $$

For the following exercises, use a calculator with CAS to answer the questions. Consider \(\frac{x^{k}}{x+1}\) with \(k=1,2,3 .\) What do you expect the result to be if \(k=4\) ?

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.For the following exercises, make a table to confirm the end behavior of the function. $$ f(x)=x^{4}-81 $$

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function. Vertex (-1,2) opens down.

For the following exercises, use the given information to answer the questions. The force exerted by the wind on a plane surface varies jointly with the square of the velocity of the wind and with the area of the plane surface. If the area of the surface is 40 square feet surface and the wind velocity is 20 miles per hour, the resulting force is 15 pounds. Find the force on a surface of 65 square feet with a velocity of 30 miles per hour.

For the following exercises, determine the function described and then use it to answer the question. The volume, \(V\), of a sphere in terms of its radius, \(r\), is given by \(V(r)=\frac{4}{3} \pi r^{3}\). Express \(r\) as a function of \(V\), and find the radius of a sphere with volume of 200 cubic feet.

For the following exercises, list all possible rational zeros for the functions. $$ f(x)=3 x^{3}+5 x^{2}-5 x+4 $$

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.For the following exercises, make a table to confirm the end behavior of the function. $$ f(x)=-x^{3}+x^{2}+2 x $$

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function. Vertex \((-5,11),\) opens down.

For the following exercises, use the given information to answer the questions. The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit \(45 \mathrm{hp}\) at \(100 \mathrm{rpm},\) what must the diameter be in order to transmit \(60 \mathrm{hp}\) at \(150 \mathrm{rpm} ?\)

For the following exercises, determine the function described and then use it to answer the question. The surface area, \(A,\) of a sphere in terms of its radius, \(r\), is given by \(A(r)=4 \pi r^{2}\). Express \(r\) as a function of \(A,\) and find the radius of a sphere with a surface area of 1000 square inches.

For the following exercises, list all possible rational zeros for the functions. $$ f(x)=6 x^{4}-10 x^{2}+13 x+1 $$

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.For the following exercises, make a table to confirm the end behavior of the function. $$ f(x)=x^{3}-2 x^{2}-15 x $$

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function. Vertex (-100,100) , opens up.

For the following exercises, use the given information to answer the questions. The kinetic energy \(K\) of a moving object varies jointly with its mass \(m\) and the square of its velocity \(v\). If an object weighing 40 kilograms with a velocity of 15 meters per second has a kinetic energy of 1000 joules, find the kinetic energy if the velocity is increased to 20 meters per second.

For the following exercises, determine the function described and then use it to answer the question. A container holds \(100 \mathrm{~mL}\) of a solution that is 25 \(\mathrm{mL}\) acid. If \(n \mathrm{~mL}\) of a solution that is \(60 \%\) acid is added, the function \(C(n)=\frac{25+.6 n}{100+n}\) gives the concentration, \(C\), as a function of the number of \(\mathrm{mL}\) added, \(n\). Express \(n\) as a function of \(C\) and determine the number of \(\mathrm{mL}\) that need to be added to have a solution that is \(50 \%\) acid.

For the following exercises, list all possible rational zeros for the functions. $$ f(x)=4 x^{5}-10 x^{4}+8 x^{3}+x^{2}-8 $$

For the following exercises, use synthetic division to determine the quotient involving a complex number. $$ \frac{x^{2}+1}{x-i} $$

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.For the following exercises, make a table to confirm the end behavior of the function. $$ f(x)=x^{3}-0.01 x $$

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (1,1) and has shape of \(f(x)=2 x^{2} .\) Vertex is on the \(y\) axis.

For the following exercises, determine the function described and then use it to answer the question. The period \(T,\) in seconds, of a simple pendulum as a function of its length \(l\), in feet, is given by \(T(l)=2 \pi \sqrt{\frac{l}{32.2}}\). Express \(l\) as a function of \(T\) and determine the length of a pendulum with period of 2 seconds.

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational. $$ f(x)=6 x^{3}-7 x^{2}+1 $$

For the following exercises, use synthetic division to determine the quotient involving a complex number. $$ \frac{x+1}{x+i} $$

For the following exercises, use the given information about the polynomial graph to write the equation. Degree \(5 .\) Double zero at \(x=1\), and triple zero at \(x=3\). Passes through the point (2,15) .

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or -1 . There may be more than one correct answer. The \(y\) - intercept is (0,-4) . The \(x\) - intercepts are \((-2,0),(2,0) .\) Degree is 2 End behavior: as \(x \rightarrow-\infty, f(x) \rightarrow \infty ;\) as \(x \rightarrow \infty\), \(f(x) \rightarrow \infty\)

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (-1,4) and has the shape of \(f(x)=2 x^{2}\). Vertex is on the \(y\) - axis.

For the following exercises, determine the function described and then use it to answer the question. The volume of a cylinder, \(V\), in terms of radius, \(r\), and height, \(h,\) is given by \(V=\pi r^{2} h .\) If a cylinder has a height of 6 meters, express the radius as a function of \(V\) and find the radius of a cylinder with volume of 300 cubic meters.

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational. $$ f(x)=4 x^{3}-4 x^{2}-13 x-5 $$

For the following exercises, use synthetic division to determine the quotient involving a complex number. $$ \frac{x^{2}+1}{x+i} $$

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3 . Zeros at \(x=4, x=3\), and \(x=2 . y\) -intercept \(a t\) (0,-24)

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or -1 . There may be more than one correct answer. The \(y\) - intercept is (0,9) . The \(x\) - intercepts are \((-3,0),(3,0) .\) Degree is 2 . End behavior: as \(x \rightarrow-\infty, \quad f(x) \rightarrow-\infty,\) as \(x \rightarrow \infty, f(x) \rightarrow-\infty\).

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (2,3) and has the shape of \(f(x)=3 x^{2}\). Vertex is on the \(y\) - axis.

For the following exercises, determine the function described and then use it to answer the question. The surface area, \(A,\) of a cylinder in terms of its radius, \(r,\) and height, \(h,\) is given by \(A=2 \pi r^{2}+2 \pi r h .\) If the height of the cylinder is 4 feet, express the radius as a function of \(A\) and find the radius if the surface area is 200 square feet.

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational. $$ f(x)=8 x^{3}-6 x^{2}-23 x+6 $$

For the following exercises, use synthetic division to determine the quotient involving a complex number. $$ \frac{x^{3}+1}{x-i} $$

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3 . Zeros at \(x=-3\), \(x=-2\) and \(x=1 . y\) -intercept at (0,12) .

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or -1 . There may be more than one correct answer. The \(y\) - intercept is (0,0) . The \(x\) - intercepts are \((0,0),(2,0) .\) Degree is \(3 .\) End behavior: as \(x \rightarrow-\infty, \quad f(x) \rightarrow-\infty,\) as \(x \rightarrow \infty, f(x) \rightarrow \infty\)

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (1,-3) and has the shape of \(f(x)=-x^{2}\). Vertex is on the \(y\) - axis.

For the following exercises, determine the function described and then use it to answer the question. The volume of a right circular cone, \(V\), in terms of its radius, \(r,\) and its height, \(h,\) is given by \(V=\frac{1}{3} \pi r^{2} h .\) Express \(r\) in terms of \(V\) if the height of the cone is 12 feet and find the radius of a cone with volume of 50 cubic inches.

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational. $$ f(x)=12 x^{4}+55 x^{3}+12 x^{2}-117 x+54 $$

For the following exercises, use the given length and area of a rectangle to express the width algebraically. Length is \(x+5,\) area is \(2 x^{2}+9 x-5\)

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 5. Roots of multiplicity 2 at \(x=-3\) and \(x=2\) and a root of multiplicity 1 at \(x=-2\). \(y\) -intercept at (0,4) .

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or -1 . There may be more than one correct answer. The \(y\) - intercept is (0,1) . The \(x\) - intercept is (1,0) . Degree is \(3 .\) End behavior: as \(x \rightarrow-\infty, \quad f(x) \rightarrow \infty,\) as \(x \rightarrow \infty, f(x) \rightarrow-\infty\)

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (4,3) and has the shape of \(f(x)=5 x^{2}\). Vertex is on the \(y\) - axis.

For the following exercises, determine the function described and then use it to answer the question. Consider a cone with height of 30 feet. Express the radius, \(r,\) in terms of the volume, \(V,\) and find the radius of a cone with volume of 1000 cubic feet.

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational. $$ f(x)=16 x^{4}-24 x^{3}+x^{2}-15 x+25 $$

For the following exercises, use the given length and area of a rectangle to express the width algebraically. Length is \(2 x+5\), area is $$ 4 x^{3}+10 x^{2}+6 x+15 $$

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 4. Roots of multiplicity 2 at \(x=\frac{1}{2}\) and roots of multiplicity 1 at \(x=6\) and \(x=-2\). \(y\) -intercept at (0,18) .

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or -1 . There may be more than one correct answer. The \(y\) - intercept is (0,1) . There is no \(x\) - intercept. Degree is 4. End behavior: as \(x \rightarrow-\infty, \quad f(x) \rightarrow \infty,\) as \(x \rightarrow \infty, f(x) \rightarrow \infty .\)

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (1,-6) has the shape of \(f(x)=3 x^{2}\). Vertex has \(\mathrm{x}\) coordinate of -1.