In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine

The graphs of all polynomial functions are ________, which means that the graphs have no breaks, holes, or gaps.

The ________ ________ ________ is used to determine the left-hand and right-hand behavior of the graph of a polynomial function.

A polynomial function of degree has at most ________ real zeros and at most ________ turning points.

If is a zero of a polynomial function then the following three statements are true. (a) is a ________ of the polynomial equation (b) ________ is a factor of the polynomial (c) is an ________ of the graph

If a real zero of a polynomial function is of even multiplicity, then the graph of ________ the -axis at and if it is of odd multiplicity then the graph of ________ the -axis at

A polynomial function is written in ________ form if its terms are written in descending order of exponents from left to right.

The ________ ________ Theorem states that if is a polynomial function such that then in the interval takes on every value between and

In Exercises 18, match the polynomial function with its graph.[The graphs are labeled (a), (b), (c), (d), (e), (f ), (g),and (h).]

In Exercises 912, sketch the graph of and each transformation.(a) (b) (c) (d)

In Exercises 912, sketch the graph of and each transformation.(a) (b) (c) (d)

In Exercises 912, sketch the graph of and each transformation.(a) (b) (c) (d) (e) (f ) f _x_ _ _12 x_4 f _x_ _ _2x_4 _ 1 _ 2

In Exercises 912, sketch the graph of and each transformation.(a) (b) (c) (d) (e) (f)

In Exercises 1322, describe the right-hand and left-hand behavior of the graph of the polynomial function.

In Exercises 1322, describe the right-hand and left-hand behavior of the graph of the polynomial function.

In Exercises 1322, describe the right-hand and left-hand behavior of the graph of the polynomial function.

In Exercises 1322, describe the right-hand and left-hand behavior of the graph of the polynomial function.

In Exercises 1322, describe the right-hand and left-hand behavior of the graph of the polynomial function.

In Exercises 1322, describe the right-hand and left-hand behavior of the graph of the polynomial function.

In Exercises 1322, describe the right-hand and left-hand behavior of the graph of the polynomial function.

In Exercises 1322, describe the right-hand and left-hand behavior of the graph of the polynomial function.

In Exercises 1322, describe the right-hand and left-hand behavior of the graph of the polynomial function.

In Exercises 1322, describe the right-hand and left-hand behavior of the graph of the polynomial function.

In Exercises 2326, use a graphing utility to graph the functions and in the same viewing window. Zoom out sufficiently far to show that the right-hand and left-hand behaviors of and appear identical.

In Exercises 2326, use a graphing utility to graph the functions and in the same viewing window. Zoom out sufficiently far to show that the right-hand and left-hand behaviors of and appear identical.

In Exercises 2326, use a graphing utility to graph the functions and in the same viewing window. Zoom out sufficiently far to show that the right-hand and left-hand behaviors of and appear identical.

In Exercises 2326, use a graphing utility to graph the functions and in the same viewing window. Zoom out sufficiently far to show that the right-hand and left-hand behaviors of and appear identical.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 2742, (a) find all the real zeros of the polynomial function, (b) determine the multiplicity of each zero and the number of turning points of the graph of the function, and (c) use a graphing utility to graph the function and verify your answers.

In Exercises 4346, (a) use a graphing utility to graph the function, (b) use the graph to approximate any -intercepts of the graph, (c) set and solve the resulting equation, and (d) compare the results of part (c) with any -intercepts of the graph.

In Exercises 4346, (a) use a graphing utility to graph the function, (b) use the graph to approximate any -intercepts of the graph, (c) set and solve the resulting equation, and (d) compare the results of part (c) with any -intercepts of the graph.

In Exercises 4346, (a) use a graphing utility to graph the function, (b) use the graph to approximate any -intercepts of the graph, (c) set and solve the resulting equation, and (d) compare the results of part (c) with any -intercepts of the graph.

In Exercises 4346, (a) use a graphing utility to graph the function, (b) use the graph to approximate any -intercepts of the graph, (c) set and solve the resulting equation, and (d) compare the results of part (c) with any -intercepts of the graph.

In Exercises 4756, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 4756, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 4756, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 4756, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 4756, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 4756, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 4756, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 4756, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 4756, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 4756, find a polynomial function that has the given zeros. (There are many correct answers.)

In Exercises 5766, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 5766, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 5766, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 5766, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 5766, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 5766, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 5766, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 5766, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 5766, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 5766, find a polynomial of degree that has the given zero(s). (There are many correct answers.)

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 6780, sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points.

In Exercises 8184, use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero.

In Exercises 8184, use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero.

In Exercises 8184, use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero.

In Exercises 8184, use a graphing utility to graph the function. Use the zero or root feature to approximate the real zeros of the function. Then determine the multiplicity of each zero.

In Exercises 8588, use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero.Adjust the table to approximate the zeros of the function. Use the zero or root feature of a graphing utility to verify your results.

In Exercises 8588, use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero.Adjust the table to approximate the zeros of the function. Use the zero or root feature of a graphing utility to verify your results.

In Exercises 8588, use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero.Adjust the table to approximate the zeros of the function. Use the zero or root feature of a graphing utility to verify your results.

In Exercises 8588, use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero.Adjust the table to approximate the zeros of the function. Use the zero or root feature of a graphing utility to verify your results.

Numerical and Graphical Analysis An open box is to be made from a square piece of material, 36 inches on a side, by cutting equal squares with sides of length from the corners and turning up the sides (see figure). (a) Verify that the volume of the box is given by the function (b) Determine the domain of the function. (c) Use a graphing utility to create a table that shows the box height and the corresponding volumes Use the table to estimate the dimensions that will produce a maximum volume. (d) Use a graphing utility to graph and use the graph to estimate the value of for which is maximum. Compare your result with that of part (c).

Maximum Volume An open box with locking tabs is to be made from a square piece of material 24 inches on a side. This is to be done by cutting equal squares from the corners and folding along the dashed lines shown in the figure. (a) Verify that the volume of the box is given by the function (b) Determine the domain of the function (c) Sketch a graph of the function and estimate the value of for which is maximum.

Construction A roofing contractor is fabricating gutters from 12-inch aluminum sheeting. The contractor plans to use an aluminum siding folding press to create the gutter by creasing equal lengths for the sidewalls (see figure). (a) Let represent the height of the sidewall of the gutter. Write a function that represents the cross-sectional area of the gutter. (b) The length of the aluminum sheeting is 16 feet. Write a function that represents the volume of one run of gutter in terms of (c) Determine the domain of the function in part (b). (d) Use a graphing utility to create a table that shows the sidewall height and the corresponding volumes Use the table to estimate the dimensions that will produce a maximum volume. (e) Use a graphing utility to graph Use the graph to estimate the value of for which is a maximum. Compare your result with that of part (d). (f ) Would the value of change if the aluminum sheeting were of different lengths? Explain.

Construction An industrial propane tank is formed by adjoining two hemispheres to the ends of a right circular cylinder. The length of the cylindrical portion of the tank is four times the radius of the hemispherical components (see figure). (a) Write a function that represents the total volume of the tank in terms of (b) Find the domain of the function. (c) Use a graphing utility to graph the function. (d) The total volume of the tank is to be 120 cubic feet. Use the graph from part (c) to estimate the radius and length of the cylindrical portion of the tank. Data Analysis: Home Prices (Source: U.S. Census Bureau; U.S. Department of Housing and Urban Development)

Use a graphing utility to plot the data and graph the model for in the same viewing window. How closely does the model represent the data?

Use a graphing utility to plot the data and graph the model for in the same viewing window. How closely does the model represent the data?

Use the models to predict the median prices of a new privately owned home in both regions in 2008. Do your answers seem reasonable? Explain.

Use the graphs of the models in Exercises 93 and 94 to write a short paragraph about the relationship between the median prices of homes in the two regions.

Tree Growth The growth of a red oak tree is approximatedby the functionwhere is the height of the tree (in feet) andis its age (in years).(a) Use a graphing utility to graph the function. (Hint:Use a viewing window in whichand(b) Estimate the age of the tree when it is growingmost rapidly. This point is called the point ofdiminishing returns because the increase in sizewill be less with each additional year.(c) Using calculus, the point of diminishing returns canalso be found by finding the vertex of the parabolagiven byFind the vertex of this parabola.(d) Compare your results from parts (b) and (c).

Revenue The total revenue (in millions of dollars) for a company is related to its advertising expense by the function where is the amount spent on advertising (in tens of thousands of dollars). Use the graph of this function, shown in the figure, to estimate the point on the graph at which the function is increasing most rapidly. This point is called the point of diminishing returns because any expense above this amount will yield less return per dollar invested in advertising.

A fifth-degree polynomial can have five turning points in its graph.

It is possible for a sixth-degree polynomial to have only one solution.

The graph of the function given by rises to the left and falls to the right.

Graphical Analysis For each graph, describe a polynomial function that could represent the graph. (Indicate the degree of the function and the sign of its leading coefficient.) (a) (b) (c) (d)

Graphical Reasoning Sketch a graph of the function given by Explain how the graph of each function differs (if it does) from the graph of each function Determine whether is odd, even, or neither. (a) (b) (c) (d) (e) (f ) (g) (h)

Exploration Explore the transformations of the form (a) Use a graphing utility to graph the functions given by and Determine whether the graphs are increasing or decreasing. Explain. (b) Will the graph of always be increasing or decreasing? If so, is this behavior determined by or Explain. (c) Use a graphing utility to graph the function given by Use the graph and the result of part (b) to determine whether can be written in the form Explain.

In Exercises 105108, factor the expression completely.

In Exercises 105108, factor the expression completely.

In Exercises 105108, factor the expression completely.

In Exercises 105108, factor the expression completely.

In Exercises 109112, solve the equation by factoring.

In Exercises 109112, solve the equation by factoring.

In Exercises 109112, solve the equation by factoring.

In Exercises 109112, solve the equation by factoring.

In Exercises 113116, solve the equation by completing the square.

In Exercises 113116, solve the equation by completing the square.

In Exercises 113116, solve the equation by completing the square.

In Exercises 113116, solve the equation by completing the square.

In Exercises 117122, describe the transformation from a common function that occurs in Then sketch its graph.

In Exercises 117122, describe the transformation from a common function that occurs in Then sketch its graph.

In Exercises 117122, describe the transformation from a common function that occurs in Then sketch its graph.

In Exercises 117122, describe the transformation from a common function that occurs in Then sketch its graph.