Chapter 3

Determine, without graphing, whether the given quadratic function has a minimum value or \(a\) maximum value. Then find the coordinates of the minimum or the maximum point. \(f(x)=3 x^{2}-12 x-1\)

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=x^{3}+x^{2}-4 x-4$$

Use the four-step procedure for solving variation problems given on page 356 to solve. Do you still own records, or are you strictly a CD person? Record owners claim that the quality of sound on good vinyl surpasses that of a CD, although this is up for debate. This, however, is not debatable: The number of revolutions a record makes as it is being played is directly proportional to the time that it is on the turntable. A record that lasted 3 minutes made 135 revolutions. If a record takes 2.4 minutes to play, how many revolutions does it make?

In Exercises \(29-36,\) find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. \(n=4 ;-4, \frac{1}{3},\) and \(2+3 i\) are zeros; \(f(1)=100\)

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ x^{4}-x^{3}+2 x^{2}-4 x-8=0 $$

Find the horizontal asymptote, if any, of the graph of each rational function. $$f(x)=\frac{-3 x+7}{5 x-2}$$

Given \(f(x)=3 x^{4}+6 x^{3}-2 x+4,\) use the Remainder Theorem to find \(f(-4)\).

Determine, without graphing, whether the given quadratic function has a minimum value or \(a\) maximum value. Then find the coordinates of the minimum or the maximum point. \(f(x)=2 x^{2}-8 x-3\)

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=x^{4}-9 x^{2}$$

In Exercises \(37-44,\) find all the zeros of the function and write the polynomial as a product of linear factors. $$ f(x)=x^{3}-x^{2}+25 x-25 $$

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ f(x)=3 x^{4}-11 x^{3}-x^{2}+19 x+6 $$

Use synthetic division to divide \(f(x)=x^{3}-4 x^{2}+\) \(x+6\) by \(x+1 .\) Use the result to find all zeros of \(f\).

Determine, without graphing, whether the given quadratic function has a minimum value or \(a\) maximum value. Then find the coordinates of the minimum or the maximum point. \(f(x)=-4 x^{2}+8 x-3\)

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=x^{4}-x^{2}$$

In Exercises \(37-44,\) find all the zeros of the function and write the polynomial as a product of linear factors. $$ f(x)=x^{3}-10 x^{2}+33 x-34 $$

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ f(x)=2 x^{4}+3 x^{3}-11 x^{2}-9 x+15 $$

Use synthetic division to divide \(f(x)=x^{3}-2 x^{2}\) \(x+2\) by \(x+1 .\) Use the result to find all zeros of \(f\).

Determine, without graphing, whether the given quadratic function has a minimum value or \(a\) maximum value. Then find the coordinates of the minimum or the maximum point. \(f(x)=-2 x^{2}-12 x+3\)

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-x^{4}+16 x^{2}$$

Use the four-step procedure for solving variation problems given on page 356 to solve. The time that it takes you to get to campus varies inversely as your driving rate. Averaging 20 miles per hour in terrible traffic, it takes you 1.5 hours to get to campus. How long would the trip take averaging 60 miles per hour?

In Exercises \(37-44,\) find all the zeros of the function and write the polynomial as a product of linear factors. $$ f(x)=x^{3}-8 x^{2}+25 x-26 $$

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ 4 x^{4}-x^{3}+5 x^{2}-2 x-6=0 $$

Solve the equation \(2 x^{3}-5 x^{2}+x+2=0\) given that 2 is a zero of \(f(x)=2 x^{3}-5 x^{2}+x+2\).

Determine, without graphing, whether the given quadratic function has a minimum value or \(a\) maximum value. Then find the coordinates of the minimum or the maximum point. \(f(x)=5 x^{2}-5 x\)

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-x^{4}+4 x^{2}$$

In Exercises \(37-44,\) find all the zeros of the function and write the polynomial as a product of linear factors. $$ f(x)=x^{3}-8 x^{2}+17 x-4 $$

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ 3 x^{4}-11 x^{3}-3 x^{2}-6 x+8=0 $$

Solve the equation \(2 x^{3}-3 x^{2}-11 x+6=0\) given that \(-2\) is a zero of \(f(x)=2 x^{3}-3 x^{2}-11 x+6\).

Determine, without graphing, whether the given quadratic function has a minimum value or \(a\) maximum value. Then find the coordinates of the minimum or the maximum point. \(f(x)=6 x^{2}-6 x\)

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=x^{4}-2 x^{3}+x^{2}$$

In Exercises \(37-44,\) find all the zeros of the function and write the polynomial as a product of linear factors. $$ f(x)=x^{4}+37 x^{2}+36 $$

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ 2 x^{5}+7 x^{4}-18 x^{2}-8 x+8=0 $$

Solve the equation \(12 x^{3}+16 x^{2}-5 x-3=0\) given that \(-\frac{3}{2}\) is a root.

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=x^{4}-6 x^{3}+9 x^{2}$$

In Exercises \(37-44,\) find all the zeros of the function and write the polynomial as a product of linear factors. $$ f(x)=x^{4}+8 x^{3}+9 x^{2}-10 x+100 $$

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. $$ 4 x^{5}+12 x^{4}-41 x^{3}-99 x^{2}+10 x+24=0 $$

Solve the equation \(3 x^{3}+7 x^{2}-22 x-8=0\) given that \(-\frac{1}{3}\) is a root.

The function $$f(x)=104.5 x^{2}-1501.5 x+6016$$ models the death rate per year per \(100,000\) males, \(f(x),\) for U.S. men who average \(x\) hours of sleep each night. How many hours of sleep, to the nearest tenth of an hour, corresponds to the minimum death rate? What is this minimum death rate, to the nearest whole number?

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-2 x^{4}+4 x^{3}$$

Use the four-step procedure for solving variation problems given on page 356 to solve. A person's body-mass index is used to assess levels of fatness, with an index from 20 to 26 considered in the desirable range. The index varies directly as one's weight, in pounds, and inversely as one's height, in inches. A person who weighs 150 pounds and is 70 inches tall has an index of 21\. What is the body-mass index of a person who weighs 240 pounds and is 74 inches tall? Because the index is rounded to the nearest whole number, do so and then determine if this person's level of fatness is in the desirable range.

In Exercises \(37-44,\) find all the zeros of the function and write the polynomial as a product of linear factors. $$ f(x)=16 x^{4}+36 x^{3}+16 x^{2}+x-30 $$

A rectangle with length \(2 x+5\) inches has an area of \(2 x^{4}+15 x^{3}+7 x^{2}-135 x-225\) square inches. Write a polynomial that represents its width.

Fireworks are launched into the air. The quadratic function $$s(t)=-16 t^{2}+200 t+4$$ models the fireworks' height, \(s(t),\) in feet, \(t\) seconds after they are launched. When should the fireworks explode so that they go off at the greatest height? What is that height?

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-2 x^{4}+2 x^{3}$$

Use the four-step procedure for solving variation problems given on page 356 to solve. The volume of a gas varies directly as its temperature and inversely as its pressure. At a temperature of 100 Kelvin and a pressure of 15 kilograms per square meter, the gas occupies a volume of 20 cubic meters. Find the volume at a temperature of 150 Kelvin and a pressure of 30 kilograms per square meter.

In Exercises \(37-44,\) find all the zeros of the function and write the polynomial as a product of linear factors. $$ f(x)=2 x^{4}-x^{3}+7 x^{2}-4 x-4 $$

A football is thrown by a quarterback to a receiver 40 yards away. The quadratic function $$s(t)=-0.025 t^{2}+t+5$$ models the football's height above the ground, s(t), in feet, when it is \(t\) yards from the quarterback. How many yards from the quarterback does the football reach its greatest height? What is that height?

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=6 x^{3}-9 x-x^{5}$$

Use the four-step procedure for solving variation problems given on page 356 to solve. The intensity of illumination on a surface varies inversely as the square of the distance of the light source from the surface. The illumination from a source is 25 foot-candles at a distance of 4 feet. What is the illumination when the distance is 6 feet?

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=6 x-x^{3}-x^{5}$$