Chapter 3

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c .\) Use your calculator to support your results. Vertex \((-2,-3) ;\) through \((0,-19)\)

RELATING CONCEPTS For individual or group investigation (Exercises \(55-60\) ) The close relationships among \(x\) -intercepts of a graph of a function, real zeros of the function, and real solutions of the associated equation should, by now, be apparent to you. Consider the graph of the polynomial function \(P(x)=x^{3}-2 x^{2}-11 x+12,\). What are the zeros of the function \(P ?\)

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c .\) Use your calculator to support your results. Vertex \((8,3) ;\) through \((10,5)\)

Multiply as indicated. Write each product in standard form. $$(2+4 i)(-1+3 i)$$

For each polynomial function, (a) list all possible rational zeros, (b) use a graph to eliminate some of the possible zeros listed in part (a), (c) find all rational zeros, and (d) factor \(P(x)\). $$P(x)=6 x^{3}+17 x^{2}-31 x-12$$

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{3}=-1$$

Solve each equation. For equations with real solutions, support your answers graphically. $$-3 x^{2}+4 x+4=0$$

RELATING CONCEPTS For individual or group investigation (Exercises \(55-60\) ) The close relationships among \(x\) -intercepts of a graph of a function, real zeros of the function, and real solutions of the associated equation should, by now, be apparent to you. Consider the graph of the polynomial function \(P(x)=x^{3}-2 x^{2}-11 x+12,\). If \(P(x)\) is divided by \(x-2,\) what is the remainder? What is \(P(2) ?\)

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c .\) Use your calculator to support your results. Vertex \((-6,-12)\); through \((6,24)\)

Multiply as indicated. Write each product in standard form. $$(1+3 i)(2-5 i)$$

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{3}=-8$$

The concepts of stretching, shrinking, translating, and reflecting graphs presented in Sections 2.2 and 2.3 can be applied to polynomial functions of the form \(P(x)=x^{n} .\) For example, the graph of \(y=-2(x+4)^{4}-6\) can be obtained from the graph of \(y=x^{4}\) by shifting 4 units to the left, stretching vertically by applying a factor of \(2,\) reflecting across the \(x\) -axis, and shifting downward 6 units, so the graph should resemble the graph at the right. If we expand the expression \(-2(x+4)^{4}-6\) algebraically, we get $$ -2 x^{4}-32 x^{3}-192 x^{2}-512 x-518 $$ Thus, the graph of \(y=-2(x+4)^{4}-6\) is the same as that of $$ y=-2 x^{4}-32 x^{3}-192 x^{2}-512 x-518 $$ In Exercises \(55-58,\) two forms of the same polynomial function are given. Sketch by hand the general shape of the graph of the function, using the concepts of Chapter \(2,\) and describe the transformations. Then, support your answer by graphing it on your calculator in a suitable window. $$\begin{array}{l} y=0.5(x-1)^{5}+13 \\ y=0.5 x^{5}-2.5 x^{4}+5 x^{3}-5 x^{2}+2.5 x+12.5 \end{array}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$-5 x^{2}+28 x+12=0$$

Use the rational zeros theorem to factor \(P(x)\). $$P(x)=12 x^{3}+20 x^{2}-x-6$$

RELATING CONCEPTS For individual or group investigation (Exercises \(55-60\) ) The close relationships among \(x\) -intercepts of a graph of a function, real zeros of the function, and real solutions of the associated equation should, by now, be apparent to you. Consider the graph of the polynomial function \(P(x)=x^{3}-2 x^{2}-11 x+12,\). Give the solution set of \(P(x)>0,\) using interval notation.

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c .\) Use your calculator to support your results. Vertex \((-4,-2) ;\) through \((2,-26)\)

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

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

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. $$P(x)=-2 x^{3}-14 x^{2}+2 x+84$$

Solve each equation. For equations with real solutions, support your answers graphically. $$(x+5)(x-6)=(2 x-1)(x-4)$$

Use the rational zeros theorem to factor \(P(x)\). $$P(x)=12 x^{3}+40 x^{2}+41 x+12$$

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c .\) Use your calculator to support your results. Vertex \((5,6) ;\) through \((1,-6)\)

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

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{3}=64$$

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. $$P(x)=-3 x^{3}+6 x^{2}+39 x-60$$

Solve each equation. For equations with real solutions, support your answers graphically. $$(x+2)(3 x-4)=(x+5)(2 x-5)$$

Use the rational zeros theorem to factor \(P(x)\). $$P(x)=24 x^{3}+40 x^{2}-2 x-12$$

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=x^{3}-2 x^{2}-5 x+6 ; \quad 3$$

Solve each problem. Heart Rate An athlete's heart rate \(R\) in beats per minute after \(x\) minutes is given by $$R(x)=2(x-4)^{2}+90$$ where \(0 \leq x \leq 8\) (a) Describe the heart rate during this period of time. (b) Determine the minimum heart rate during this 8 -minute period.

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

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{4}=16$$

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. $$P(x)=x^{5}+4 x^{4}-3 x^{3}-17 x^{2}+6 x+9$$

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

Use the rational zeros theorem to factor \(P(x)\). $$P(x)=24 x^{3}+80 x^{2}+82 x+24$$

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=x^{3}+2 x^{2}-11 x-12 ; \quad 3$$

Solve each problem. Social Security Assets The graph shows how Social Security assets are expected to change as the number of retirees receiving benefits increases. (BAR GRAPH CAN'T COPY) The graph suggests that a quadratic function would be a good fit to the data, which are approximated by $$f(x)=-10.36 x^{2}+431.8 x-650$$ In the model, \(x=10\) represents \(2010, x=15\) represents 2015 and so on, and \(f(x)\) is in billions of dollars. (a) Explain why the coefficient of \(x^{2}\) in the model is negative, based on the graph. (b) Analytically determine the vertex of the graph. (c) Interpret the answer to part (b) as it relates to this application.

Multiply as indicated. Write each product in standard form. $$(-5-i)(5+i)$$

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{4}=81$$

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. $$P(x)=-2 x^{5}+7 x^{4}+x^{3}-20 x^{2}+4 x+16$$

Solve each quadratic equation by completing the square. $$x^{2}+2 x=4$$

Find all rational zeros of each polynomial function. $$P(x)=x^{3}+\frac{1}{2} x^{2}-\frac{11}{2} x-5$$

For each polynomial, at least one zero is given. Find all others analytically. $$P(x)=x^{3}-2 x+1 ; 1$$

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

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{3}=-64$$

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. $$P(x)=2 x^{4}+3 x^{3}-17 x^{2}-6 x-72$$

Solve each quadratic equation by completing the square. $$2 x^{2}+6 x-3=0$$

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

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

Multiply as indicated. Write each product in standard form. $$(6-4 i)(6+4 i)$$

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