Chapter 3

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. $$2 x^{2}-4 x+1=0$$

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

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{5}$$

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.

The monthly average high temperatures in degrees Fahrenheit at Daytona Beach can be modeled by $$ P(x)=0.0145 x^{4}-0.426 x^{3}+3.53 x^{2}-6.23 x+72 $$ where \(x=1\) corresponds to January and \(x=12\) represents December. (a) Find the average high temperature during March and July. (b) Estimate graphically and numerically the months when the average high temperature is about \(80^{\circ} \mathrm{F}\).

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. $$9 x^{2}+11 x+4=0$$

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

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

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{8}$$

A storage tank for butane gas is to be built in the shape of a right circular cylinder having altitude 12 feet, as shown, with a half sphere attached to each end. If \(x\) represents the radius of each half sphere, what radius should be used to cause the volume of the tank to be \(144 \pi\) cubic feet?

Heating costs In colder climates the cost for natural gas to heat homes can vary from one month to the next. The polynomial function f(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 \(x=12\) to December. (Source: Minnegasco.) (a) Where might the absolute extrema occur for \(1 \leq x \leq 12 ?\) \(\Rightarrow\) (b) Approximate the absolute extrema and interpret the results.

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. $$3 x^{2}=4 x-5$$

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

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

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{15}$$

Volume of a Box \(\quad\) 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 corner 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 your 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.

Concept Check For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverse.) $$4,5$$

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

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

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{19}$$

Highway Design To allow enough distance for cars to pass on two-lane highways, engineers calculate minimum sight distances between curves and hills. The table at top of the next page shows the minimum sight distance \(y\) in feet for a car traveling at \(x\) mph. (Image can't copy) $$\begin{array}{|c|c|c|c|c|} \hline x \text { (in mph) } & 20 & 30 & 40 & 50 \\ \hline y \text { (in feet) } & 810 & 1090 & 1480 & 1840 \end{array}$$ $$\begin{array}{|c|c|c|c} \hline x \text { (in mph) } & 60 & 65 & 70 \\ \hline y \text { (in feet) } & 2140 & 2310 & 2490 \end{array}$$ (a) Make a scatter diagram of the data. (b) Use the regression feature of a calculator to find the best-fitting linear function for the data. Graph the function with the data. (c) Repeat part (b) for a cubic function. (d) Use both functions from parts (b) and (c) to estimate the minimum sight distance for a car traveling 43 mph. (e) Which function fits the data better?

Concept Check For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverse.) $$-3,2$$

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 \(P\). $$P(x)=x^{3}+5 x^{2}-3 x-15 ; \quad k=-5$$

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{64}$$

Water Pollution Copper in high doses can be lethal to aquatic life. The table lists copper concentrations in mussels after 45 days at various distances downstream from an electroplating plant. The concentration \(C\) is measured in micrograms of copper per gram of mussel \(x\) kilometers downstream. See the table at the top of the next column. $$\begin{array}{c|c|c|c|c|c} \hline x & 5 & 21 & 37 & 53 & 59 \\ \hline C & 20 & 13 & 9 & 6 & 5 \end{array}$$ (a) Make a scatter diagram of the data. (b) Use the regression feature of a calculator to find the best-fitting quadratic function \(C\) for the data. Graph the function with the data. (c) Repeat part (b) for a cubic function. (d) By comparing graphs of the functions in parts (b) and (c) with the data, decide which function best fits the given data. (e) Concentrations above 10 are lethal to mussels. Use the cubic function to find the values of \(x\) to the nearest hundredth for which this is the case.

Concept Check For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverse.) $$1+\sqrt{2}, 1-\sqrt{2}$$

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 \(P\). $$P(x)=x^{3}+9 x^{2}-7 x-63 ; \quad k=-9$$

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{102}$$

Concept Check For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverse.) $$i,-i$$

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. $$P(x)=5 x^{4}+3 x^{2}+2 x-9$$

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

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{-6}$$

Concept Check For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverse.) $$2 i,-2 i$$

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. $$P(x)=3 x^{4}+2 x^{3}-8 x^{2}-10 x-1$$

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

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$i^{-15}$$

Concept Check For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverse.) $$1+\sqrt{3}, 1-\sqrt{3}$$

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. $$P(x)=x^{5}+3 x^{4}-x^{3}+2 x+3$$

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

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$\frac{1}{i^{9}}$$

Concept Check For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverse.) $$2-\sqrt{5}, 2+\sqrt{5}$$

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. $$P(x)=2 x^{5}-x^{4}+x^{3}-x^{2}+x+5$$

Divide. $$\frac{20 x^{4}+6 x^{3}-2 x^{2}+15 x-2}{5 x-1}$$

Simplify each power of i to \(i, 1,-i,\) or \(-1\). $$\frac{1}{i^{12}}$$

Concept Check For each pair of numbers, find the values of \(a, b,\) and \(c\) for which the quadratic equation ax \(^{2}+b x+c=0\) has the given numbers as solutions. Answers may vary. (Hint: Use the zero-product property in reverse.) $$3 i,-3 i$$

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=x^{4}-x^{3}+3 x^{2}-8 x+8\); no real zero greater than 2

Divide. $$\frac{5 x^{4}-2 x^{2}+6}{x^{2}+2}$$

Sketch a graph of \(f(x)=a x^{2}+b x+c\) that satisfies each set of conditions. $$a<0, b^{2}-4 a c=0$$