Chapter 3

In Exercises \(25-32,\) 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 $$ n-3 ; 6 \text { and }-5+2 i \text { are zeros; } f(2)--636 $$

Find the zeros for each polynomial function and give the multiplicity for each zera. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zera $$ f(x)=3\left(x+\frac{1}{2}\right)(x-4)^{3} $$

Divide using synthetic division. $$ \frac{x^{7}+x^{5}-10 x^{3}+12}{x+2} $$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-x^{2}-2 x-15$$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ (x-1)(x-2)(x-3) \geq 0 $$

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$f(x)=\frac{x^{2}-9}{x-3}$$

Find the zeros for each polynomial function and give the multiplicity for each zera. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zera $$ f(x)=x^{3}-2 x^{2}+x $$

Divide using synthetic division. $$ \frac{x^{4}-256}{x-4} $$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-x^{2}+3 x-10$$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ (x+1)(x+2)(x+3) \geq 0 $$

Use the four-step procedure for solving variation problems given on page 424 to solve. The illumination provided by a car's headlight varies inversely as the square of the distance from the headlight. A car's headlight produces an illumination of 3.75 footcandles at a distance of 40 feet. What is the illumination when the distance is 50 feet?

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$f(x)=\frac{x^{2}-25}{x-5}$$

Find the zeros for each polynomial function and give the multiplicity for each zera. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zera $$ f(x)=x^{3}+4 x^{2}+4 x $$

Divide using synthetic division. $$ \frac{x^{7}-128}{x-2} $$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-2 x^{2}-7 x-4$$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x(3-x)(x-5) \leq 0 $$

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$g(x)=\frac{x-3}{x^{2}-9}$$

In Exercises \(25-32,\) 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 \(n-4 ;-2,5,\) and \(3+2 i\) are zeros; \(f(1)--96\)

Find the zeros for each polynomial function and give the multiplicity for each zera. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zera $$ f(x)=x^{3}+7 x^{2}-4 x-28 $$

Divide using synthetic division. $$ \frac{2 x^{5}-3 x^{4}+x^{3}-x^{2}+2 x-1}{x+2} $$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x(4-x)(x-6) \leq 0 $$

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$g(x)=\frac{x-5}{x^{2}-25}$$

Find the zeros for each polynomial function and give the multiplicity for each zera. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zera $$ f(x)=x^{3}+5 x^{2}-9 x-45 $$

Divide using synthetic division. $$ \frac{x^{5}-2 x^{4}-x^{3}+3 x^{2}-x+1}{x-2} $$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-5-4 x-x^{2}$$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ (2-x)^{2}\left(x-\frac{7}{2}\right)<0 $$

Use the four-step procedure for solving variation problems given on page 424 to solve. The heat loss of a glass window varies jointly as the window's area and the difference between the outside and inside temperatures. A window 3 feet wide by 6 feet long loses 1200 Btu per hour when the temperature outside is \(20^{\circ}\) colder than the temperature inside. Find the heat loss through a glass window that is 6 feet wide by 9 feet long when the temperature outside is \(10^{\circ}\) colder than the temperature inside.

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$h(x)=\frac{x+7}{x^{2}+4 x-21}$$

In Exercises \(33-38,\) use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)-x^{3}+2 x^{2}+5 x+4 $$

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$ f(x)=x^{3}-x-1 ; \text { between } 1 \text { and } 2 $$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-x^{2}+6 x+3$$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ (5-x)^{2}\left(x-\frac{13}{2}\right)<0 $$

Use the four-step procedure for solving variation problems given on page 424 to solve. Kinetic energy varies jointly as the mass and the square of the velocity. A mass of 8 grams and velocity of 3 centimeters per second has a kinetic energy of 36 ergs. Find the kinetic energy for a mass of 4 grams and velocity of 6 centimeters per second.

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$h(x)=\frac{x+6}{x^{2}+2 x-24}$$

In Exercises \(33-38,\) use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)-x^{3}+7 x^{2}+x+7 $$

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$ f(x)=x^{3}-4 x^{2}+2 ; \text { between } 0 \text { and } 1 $$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-x^{2}+4 x-1$$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3}+2 x^{2}-x-2 \geq 0 $$

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$r(x)=\frac{x^{2}+4 x-21}{x+7}$$

In Exercises \(33-38,\) use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)-5 x^{3}-3 x^{2}+3 x-1 $$

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$ f(x)=2 x^{4}-4 x^{2}+1 ; \text { between }-1 \text { and } 0 $$

Use synthetic division and the Remainder Theorem to find the indicated function value. $$ f(x)=3 x^{3}-7 x^{2}-2 x+5 ; f(-3) $$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-2 x^{2}+4 x-3$$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3}+2 x^{2}-4 x-8 \geq 0 $$

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$r(x)=\frac{x^{2}+2 x-24}{x+6}$$

In Exercises \(33-38,\) use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)--2 x^{3}+x^{2}-x+7 $$

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$ f(x)=x^{4}+6 x^{3}-18 x^{2} ; \text { between } 2 \text { and } 3 $$

Use synthetic division and the Remainder Theorem to find the indicated function value. $$ f(x)=4 x^{3}+5 x^{2}-6 x-4 ; \quad f(-2) $$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-3 x^{2}-2 x-4$$

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3}-3 x^{2}-9 x+27<0 $$