Chapter 3

For the functions in Exercises \(59-66,\) use your graphing calculator to find a comprehensive graph and answer each of the following. (a) Determine the domain. (b) Determine all local minimum points, and tell if any is an absolute minimum point. (Approximate coordinates to the nearest hundredth.) (c) Determine all local maximum points, and tell if any is an absolute maximum point. (Approximate coordinates to the nearest hundredth.) (d) Determine the range. (If an approximation is necessary, give it to the nearest hundredth.) (e) Determine all intercepts. For each function, there is at least one \(x\) -intercept that has an integer \(x\) -value. For those that are not integers, give approximations to the nearest hundredth. Determine the \(y\) -intercept analytically. (f) Give the open interval(s) over which the function is increasing. (g) Give the open interval(s) over which the function is decreasing. $$P(x)=3 x^{4}-33 x^{2}+54$$

Solve each quadratic equation by completing the square. $$3 x^{2}-3 x-1=0$$

Find all rational zeros of each polynomial function. $$P(x)=\frac{1}{6} x^{4}-\frac{11}{12} x^{3}+\frac{7}{6} x^{2}-\frac{11}{12} x+1$$

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=3 x^{3}+5 x^{2}-3 x-2 ;-2$$

The formula for the height of a projectile is $$s(t)=-16 t^{2}+v_{0} t+s_{0}$$ where \(t\) is time in seconds, \(s_{0}\) is the initial height in feet, \(v_{0}\) is the initial velocity in feet per second, and \(s(t)\) is in feet. Use this formula to solve. A rock is launched upward from ground level with an initial velocity of 90 feet per second. Let \(t\) represent the amount of time elapsed after it is launched. (a) Explain why \(t\) cannot be a negative number in this situation. (b) Explain why \(s_{0}=0\) in this problem. (c) Give the function \(s\) that describes the height of the rock as a function of \(t\) (d) How high will the rock be 1.5 seconds after it is launched? (e) What is the maximum height attained by the rock? After how many seconds will this happen? Determine the answer analytically and graphically. (f) After how many seconds will the rock hit the ground? Determine the answer graphically.

Multiply as indicated. Write each product in standard form. $$(\sqrt{6}+i)(\sqrt{6}-i)$$

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{2}=-18$$

For the functions in Exercises \(59-66,\) use your graphing calculator to find a comprehensive graph and answer each of the following. (a) Determine the domain. (b) Determine all local minimum points, and tell if any is an absolute minimum point. (Approximate coordinates to the nearest hundredth.) (c) Determine all local maximum points, and tell if any is an absolute maximum point. (Approximate coordinates to the nearest hundredth.) (d) Determine the range. (If an approximation is necessary, give it to the nearest hundredth.) (e) Determine all intercepts. For each function, there is at least one \(x\) -intercept that has an integer \(x\) -value. For those that are not integers, give approximations to the nearest hundredth. Determine the \(y\) -intercept analytically. (f) Give the open interval(s) over which the function is increasing. (g) Give the open interval(s) over which the function is decreasing. $$P(x)=-x^{6}+24 x^{4}-144 x^{2}+256$$

Solve each quadratic equation by completing the square. $$x(x-1)=3$$

Find all rational zeros of each polynomial function. $$P(x)=x^{4}-\frac{1}{6} x^{3}+\frac{2}{3} x^{2}-\frac{1}{6} x-\frac{1}{3}$$

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=x^{3}-7 x^{2}+13 x-3 ; 3$$

The formula for the height of a projectile is $$s(t)=-16 t^{2}+v_{0} t+s_{0}$$ where \(t\) is time in seconds, \(s_{0}\) is the initial height in feet, \(v_{0}\) is the initial velocity in feet per second, and \(s(t)\) is in feet. Use this formula to solve. A toy rocket is launched from the top of a building 50 feet tall at an initial velocity of 200 feet per second. Let \(t\) represent the amount of time elapsed after the launch. (a) Express the height \(s\) as a function of the time \(t\) (b) Determine both analytically and graphically the time at which the rocket reaches its highest point. How high will it be at that time? (c) For what time interval will the rocket be more than 300 feet above ground level? Determine the answer graphically, and give times to the nearest tenth of a second. (d) After how many seconds will the rocket hit the ground? Determine the answer graphically.

Multiply as indicated. Write each product in standard form. $$(\sqrt{2}-4 i)(\sqrt{2}+4 i)$$

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{2}=-52$$

For the functions in Exercises \(59-66,\) use your graphing calculator to find a comprehensive graph and answer each of the following. (a) Determine the domain. (b) Determine all local minimum points, and tell if any is an absolute minimum point. (Approximate coordinates to the nearest hundredth.) (c) Determine all local maximum points, and tell if any is an absolute maximum point. (Approximate coordinates to the nearest hundredth.) (d) Determine the range. (If an approximation is necessary, give it to the nearest hundredth.) (e) Determine all intercepts. For each function, there is at least one \(x\) -intercept that has an integer \(x\) -value. For those that are not integers, give approximations to the nearest hundredth. Determine the \(y\) -intercept analytically. (f) Give the open interval(s) over which the function is increasing. (g) Give the open interval(s) over which the function is decreasing. $$P(x)=-3 x^{6}+2 x^{5}+9 x^{4}-8 x^{3}+11 x^{2}+4$$

Solve each quadratic equation by completing the square. $$2 x(2 x-5)=2$$

Use the rational zeros theorem to completely factor \(P(x)\). $$P(x)=6 x^{4}-5 x^{3}-11 x^{2}+10 x-2$$

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=x^{4}-41 x^{2}+180 ; \quad-6 \text { and } 6$$

The formula for the height of a projectile is $$s(t)=-16 t^{2}+v_{0} t+s_{0}$$ where \(t\) is time in seconds, \(s_{0}\) is the initial height in feet, \(v_{0}\) is the initial velocity in feet per second, and \(s(t)\) is in feet. Use this formula to solve. A ball is launched upward from ground level with an initial velocity of 150 feet per second. (a) Determine graphically whether the ball will reach a height of 355 feet. If it will, determine the time(s) when this happens. If it will not, explain why, using a graphical interpretation. (b) Repeat part (a) for a ball launched from a height of 30 feet with an initial velocity of 250 feet per second.

Multiply as indicated. Write each product in standard form. $$i(3-4 i)(3+4 i)$$

Floating Ball The polynomial function $$ f(x)=\frac{\pi}{3} x^{3}-5 \pi x^{2}+\frac{500 \pi d}{3} $$ can be used to find the depth that a ball 10 centimeters in diameter sinks in water. The constant \(d\) is the density of the ball, where the density of water is \(1 .\) The smallest positive zero of \(f(x)\) equals the depth that the ball sinks. Approximate this depth for each material and interpret the results. (a) A wooden ball with \(d=0.8\) (b) A solid aluminum ball with \(d=2.7\) (c) A spherical water balloon with \(d=1\)

Solve each quadratic equation by completing the square. $$2 x^{2}-x+3=0$$

Use the rational zeros theorem to completely factor \(P(x)\). $$P(x)=5 x^{4}+8 x^{3}-19 x^{2}-24 x+12$$

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=x^{4}-52 x^{2}+147 ; \quad-7 \text { and } 7$$

The formula for the height of a projectile is $$s(t)=-16 t^{2}+v_{0} t+s_{0}$$ where \(t\) is time in seconds, \(s_{0}\) is the initial height in feet, \(v_{0}\) is the initial velocity in feet per second, and \(s(t)\) is in feet. Use this formula to solve. An astronaut on the moon throws a baseball upward. The astronaut is 6 feet, 6 inches tall and the initial velocity of the ball is 30 feet per second. The height of the ball is approximated by the function $$s(t)=-2.7 t^{2}+30 t+6.5$$ where \(t\) is the number of seconds after the ball was thrown. (a) After how many seconds is the ball 12 feet above the moon's surface? (b) How many seconds after it is thrown will the ball return to the surface? (c) The ball will never reach a height of 100 feet. How can this be determined analytically?

Multiply as indicated. Write each product in standard form. $$i(2+7 i)(2-7 i)$$

Solve each quadratic equation by completing the square. $$x^{2}-2 x=-5$$

Use the rational zeros theorem to completely factor \(P(x)\). $$P(x)=21 x^{4}+13 x^{3}-103 x^{2}-65 x-10$$

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=-x^{3}+8 x^{2}+3 x-24 ; 8$$

Sketch a graph of a quadratic function that satisfies each set of given conditions. Use symmetry to label another point on your graph. Vertex \((-2,-3) ;\) through \((1,4)\)

Multiply as indicated. Write each product in standard form. $$3 i(2-i)^{2}$$

Volume of a Box \(\mathrm{A}\) rectangular piece of cardboard measuring 12 inches by 18 inches is to be made into a box with an open top by cutting equal-sized squares from each corner and folding up the sides. Let \(x\) represent the length of a side of each such square in inches. (Square can't copy) (a) Give the restrictions on \(x .\) (b) Determine a function \(V\) that gives the volume of the box as a function of \(x .\) (c) For what value of \(x\) will the volume be a maximum? What is this maximum volume? (d) For what values of \(x\) will the volume be greater than 80 cubic inches?

Determine a window that will provide a comprehensive graph of each polynomial function. (In each case, there are many possible such windows.) $$P(x)=2.9 x^{3}-37 x^{2}+28 x-143$$

Evaluate the discriminant, and use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not actually solve the equation. $$x^{2}+8 x+16=0$$

Use the rational zeros theorem to completely factor \(P(x)\). $$P(x)=2 x^{4}+7 x^{3}-9 x^{2}-49 x-35$$

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=-x^{3}+4 x^{2}+7 x-28 ; 4$$

Sketch a graph of a quadratic function that satisfies each set of given conditions. Use symmetry to label another point on your graph. Vertex \((5,6) ;\) through \((1,-6)\)

Multiply as indicated. Write each product in standard form. $$-5 i(4-3 i)^{2}$$

Evaluate the discriminant, and use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not actually solve the equation. $$8 x^{2}=14 x-3$$

Use the given zero to completely factor \(P(x)\) into linear factors. $$\text { Zero: } i ; P(x)=x^{5}-x^{4}+5 x^{3}-5 x^{2}+4 x-4$$

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=2 x^{3}-3 x^{2}-17 x+30 ; \quad k=2$$

Sketch a graph of a quadratic function that satisfies each set of given conditions. Use symmetry to label another point on your graph. Maximum value of 1 at \(x=3 ; y\) -intercept is \((0,-4)\)

Multiply as indicated. Write each product in standard form. $$(2+i)(2-i)(4+3 i)$$

Buoyancy of a Spherical Object It has been determined that a spherical object of radius 4 inches with specific gravity 0.25 will sink in water to a depth of \(x\) inches, where \(x\) is the least positive root of the equation $$ x^{3}-12 x^{2}+64=0 $$ To what depth will this object sink if \(x<10 ?\)

Evaluate the discriminant, and use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not actually solve the equation. $$4 x^{2}=6 x+3$$

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=2 x^{3}-3 x^{2}-5 x+6 ; \quad k=1$$

Sketch a graph of a quadratic function that satisfies each set of given conditions. Use symmetry to label another point on your graph. Minimum value of \(-4\) at \(x=-3 ; y\) -intercept is \((0,3)\)

Multiply as indicated. Write each product in standard form. $$(3-i)(3+i)(2-6 i)$$

Use the given zero to completely factor \(P(x)\) into linear factors. $$\text { Zero: }-3 i ; P(x)=x^{5}+2 x^{4}+10 x^{3}+20 x^{2}+9 x+18$$

Determine a window that will provide a comprehensive graph of each polynomial function. (In each case, there are many possible such windows.) $$P(x)=2 \pi x^{4}-12 x^{2}+100$$