Chapter 4

Use a 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. $$P(x)=3 x^{4}-33 x^{2}+54$$

For each polynomial at least one zero is given. Find all others analytically. $$P(x)=2 x^{3}+8 x^{2}-11 x-5 ;-5$$

Find all rational zeros of each polynomial function. $$P(x)=\frac{10}{7} x^{4}-x^{3}-7 x^{2}+5 x-\frac{5}{7}$$

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{3}=-27$$

Use a 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. $$P(x)=-x^{6}+24 x^{4}-144 x^{2}+256$$

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

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

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

Use a 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. $$P(x)=-3 x^{6}+2 x^{5}+9 x^{4}-8 x^{3}+11 x^{2}+4$$

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

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

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

Determine a window that will provide a comprehensive graph of each polynomial function. ( In each case, there are many possible such windows. $$P(x)=4 x^{5}-x^{3}+x^{2}+3 x-16$$

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

Use the rational zeros theorem to completely factor \(P(x)\) into linear factors. (Hint: Not all zeros of \(P(x)\) are rational. $$P(x)=6 x^{4}-5 x^{3}-11 x^{2}+10 x-2$$

Solve each problem. Give approximations of linear measures to the nearest hundredth. 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 \(d=1\)

Determine a window that will provide a comprehensive graph of each polynomial function. ( In each case, there are many possible such windows. $$P(x)=3 x^{5}-x^{4}+12 x^{2}-25$$

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

Use the rational zeros theorem to completely factor \(P(x)\) into linear factors. (Hint: Not all zeros of \(P(x)\) are rational. $$P(x)=5 x^{4}+8 x^{3}-19 x^{2}-24 x+12$$

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

Use the rational zeros theorem to completely factor \(P(x)\) into linear factors. (Hint: Not all zeros of \(P(x)\) are rational. $$P(x)=21 x^{4}+13 x^{3}-103 x^{2}-65 x-10$$

Determine a window that will provide a comprehensive graph of each polynomial function. ( In each case, there are many possible such windows. $$P(x)=-5.9 x^{3}+16 x^{2}-120$$

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

Use the rational zeros theorem to completely factor \(P(x)\) into linear factors. (Hint: Not all zeros of \(P(x)\) are rational. $$P(x)=2 x^{4}+7 x^{3}-9 x^{2}-49 x-35$$

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

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

Solve each problem. Give approximations of linear measures to the nearest hundredth. 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 \(x<10 ?\)

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

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

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

Solve each problem. Average High Temperatures. The monthly average high temperatures in degrees Fahrenheit in Detroit, Michigan, can be modeled by \(P(x)=0.0376 x^{4}-1.087 x^{3}+8.973 x^{2}-16.326 x+40.280\) where \(x=1\) corresponds to January and \(x=12\) represents December. (Source: www.currentresults.com) (a) Find the average high temperature during March and July. Round to the nearest degree. (b) Estimate graphically and numerically the months when the average high temperature is about \(80^{\circ} \mathrm{F}\).

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=6 x^{3}+25 x^{2}+3 x-4 ; \quad k=-4$$

Use the given zero to completely factor \(P(x)\) into linear factors. Zero: \(-2 i ; \quad P(x)=x^{4}+x^{3}+2 x^{2}+4 x-8\)

Solve each problem. Give approximations of linear measures to the nearest hundredth. Sides of a Right Triangle A certain right triangle has area 84 square inches. One leg of the triangle measures 1 inch less than the hypotenuse. Let \(x\) represent the length of the hypotenuse. (a) Express the length of the leg described in terms of \(x\) (b) Express the length of the other leg in terms of \(x .\) (c) Write an equation based on the information determined thus far. Square each side, and then write the equation with one side as a polynomial with integer coefficients, in descending powers, and the other side equal to 0 (d) Solve the equation in part (c) graphically. Find the lengths of the three sides of the triangle.

Solve each problem. Heating Costs In colder climates the cost for natural gas to heat homes can vary from one month to the next. The polynomial function \(P(x)=-0.1213 x^{4}+3.462 x^{3}-29.22 x^{2}+64.68 x+97.69\) models the monthly cost in dollars of heating a typical home. The input \(x\) represents the month, where \(x=1\) corresponds to January and \(x=12\) represents December. (Source: Minnegasco.) (a) Where might the absolute extrema occur for $$1 \leq x \leq 12 ?$$ (b) Approximate the absolute extrema and interpret the results.

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=8 x^{3}+50 x^{2}+47 x-15 ; \quad k=-5$$

Use the given zero to completely factor \(P(x)\) into linear factors. Zero: \(5 i ; \quad P(x)=x^{4}-x^{3}+23 x^{2}-25 x-50\)

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=-6 x^{3}-13 x^{2}+14 x-3 ; \quad k=-3$$

Use the given zero to completely factor \(P(x)\) into linear factors. Zero: \(1+i ; \quad P(x)=x^{4}-2 x^{3}+3 x^{2}-2 x+2\)

Solve each problem. Give approximations of linear measures to the nearest hundredth. Volume of a Box A standard piece of notebook paper measuring 8.5 inches by 11 inches is to be made into a box with an open top by cutting equal-sized squares from each comer and folding up the sides. Let \(x\) represent the length of a side of each such square in inches. (a) Use the table feature of a graphing calculator to find the maximum volume of the box. (b) Use the table feature to determine to the nearest hundredth when the volume of the box will be greater than 40 cubic inches.

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=-6 x^{3}-17 x^{2}+63 x-10 ; \quad k=-5$$

Use the given zero to completely factor \(P(x)\) into linear factors. Zero: \(2-i ; \quad P(x)=x^{4}-4 x^{3}+9 x^{2}-16 x+20\)

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=x^{3}+5 x^{2}-3 x-15 ; \quad k=-5$$

Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros for \(P(x) .\) Then use a graph to determine the actual numbers of positive and negative real zeros. $$P(x)=2 x^{3}-4 x^{2}+2 x+7$$

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=x^{3}+9 x^{2}-7 x-63 ; \quad k=-9$$

Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros for \(P(x) .\) Then use a graph to determine the actual numbers of positive and negative real zeros. $$P(x)=x^{3}+2 x^{2}+x-10$$

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=x^{3}-2 x^{2}-7 x-4 ; \quad k=-1$$

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=x^{3}+x^{2}-21 x-45 ; \quad k=-3$$

Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros for \(P(x) .\) Then use a graph to determine the actual numbers of positive and negative real zeros. $$P(x)=3 x^{4}+2 x^{3}-8 x^{2}-10 x-1$$

Divide. $$\frac{3 x^{4}-7 x^{3}+6 x-16}{3 x-7}$$