Chapter 4

Solve each equation and inequality, where \(k\) is a positive constant. (a) \(4 x^{3}-k x=0\) (b) \(4 x^{3}-k x>0\)

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$-0.5 ; \quad P(x)=4 x^{3}+12 x^{2}+7 x+1$$

Use the concepts of this section to answer the following. What are the possible numbers of real zeros (counting multiplicities) for a polynomial function with real coefficients of degree \(5 ?\)

Use a graphical method to find all real solutions of each equation. Express solutions to the nearest hundredth. $$0.86 x^{3}-5.24 x^{2}+3.55 x+7.84=0$$

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$-0.25 ; \quad P(x)=8 x^{3}+6 x^{2}-3 x-1$$

Use the concepts of this section to answer the following. Explain why a polynomial function of degree 4 with real coefficients has either zero, two, or four real zeros (counting multiplicities).

Use a graphical method to find all real solutions of each equation. Express solutions to the nearest hundredth. $$-2.47 x^{3}-6.58 x^{2}-3.33 x+0.14=0$$

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$-5 ; \quad P(x)=8 x^{3}+50 x^{2}+47 x+15$$

Use the concepts of this section to answer the following. Determine whether the description of the polynomial function \(P(x)\) with real coefficients is possible or not possible. (a) \(P(x)\) is of degree 3 and has zeros of \(1,2,\) and \(1+i\) (b) \(P(x)\) is of degree 4 and has four nonreal complex zeros. (c) \(P(x)\) is of degree 5 and \(-6\) is a zero of multiplicity 6 (d) \(P(x)\) has \(1+2 i\) as a zero of multiplicity 2

Use a graphical method to find all real solutions of each equation. Express solutions to the nearest hundredth. $$-\sqrt{7} x^{3}+\sqrt{5} x^{2}+\sqrt{17}=0$$

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$-4: \quad P(x)=6 x^{3}+25 x^{2}+3 x-3$$

Use a graphical method to find all real solutions of each equation. Express solutions to the nearest hundredth. $$\sqrt{10} x^{3}-\sqrt{11} x-\sqrt{8}=0$$

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$\sqrt{6} ; \quad P(x)=-2 x^{6}+5 x^{4}-3 x^{2}+270$$

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 \(P(x)\). $$P(x)=x^{3}-2 x^{2}-13 x-10$$

Use a graphical method to find all real solutions of each equation. Express solutions to the nearest hundredth. $$2.45 x^{4}-3.22 x^{3}=-0.47 x^{2}+6.54 x+3$$

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$\sqrt{7} ; \quad P(x)=-3 x^{6}+7 x^{4}-5 x^{2}+721$$

Use a graphical method to find all real solutions of each equation. Express solutions to the nearest hundredth. $$\sqrt{17} x^{4}-\sqrt{22} x^{2}=-1$$

RELATING CONCEPTS For individual or group investigation (Exercises \(55-58\) ) The concepts of stretching, shrinking, translating, and reflecting graphs presented earlier 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 downwand 6 units, so the graph should resemble the graph to the right. If we expand the expression \(-2(x+4)^{4}-6\) algebraically, we obtain $$ -2 x^{4}-32 x^{3}-192 x^{2}-512 x-518 $$ 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 finction are given. Sketch by hand the general shape of the graph of the function and describe the transformations. Then support your answer by graphing it on your calculator in a suitable window. $$\begin{aligned} &y=2(x+3)^{4}-7\\\ &y=2 x^{4}+24 x^{3}+108 x^{2}+216 x+155 \end{aligned}$$

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 \(P(x)\). $$P(x)=x^{3}+6 x^{2}-x-30$$

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{2}=-1$$

RELATING CONCEPTS For individual or group investigation (Exercises \(55-58\) ) The concepts of stretching, shrinking, translating, and reflecting graphs presented earlier 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 downwand 6 units, so the graph should resemble the graph to the right. If we expand the expression \(-2(x+4)^{4}-6\) algebraically, we obtain $$ -2 x^{4}-32 x^{3}-192 x^{2}-512 x-518 $$ 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 finction are given. Sketch by hand the general shape of the graph of the function and describe the transformations. Then support your answer by graphing it on your calculator in a suitable window. $$\begin{array}{l} y=-3(x+1)^{4}+12 \\ y=-3 x^{4}-12 x^{3}-18 x^{2}-12 x+9 \end{array}$$

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 \(P\), given by \(Y_{1}=X^{3}-2 X^{2}-11 X+12.\) (GRAPH CANT COPY) What are the zeros of the function \(P ?\)

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 \(P(x)\). $$P(x)=x^{3}-x^{2}-10 x-8$$

Find all \(n\) complex solutions of each equation of the form \(x^{n}=k\) $$x^{2}=-4$$

RELATING CONCEPTS For individual or group investigation (Exercises \(55-58\) ) The concepts of stretching, shrinking, translating, and reflecting graphs presented earlier 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 downwand 6 units, so the graph should resemble the graph to the right. If we expand the expression \(-2(x+4)^{4}-6\) algebraically, we obtain $$ -2 x^{4}-32 x^{3}-192 x^{2}-512 x-518 $$ 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 finction are given. Sketch by hand the general shape of the graph of the function and describe the transformations. Then support your answer by graphing it on your calculator in a suitable window. $$\begin{array}{l}y=-3(x-1)^{3}+12 \\\y=-3 x^{3}+9 x^{2}-9 x+15\end{array}$$

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 \(P(x)\). $$P(x)=6 x^{3}+17 x^{2}-31 x-12$$

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 \(P\), given by \(Y_{1}=X^{3}-2 X^{2}-11 X+12.\) (GRAPH CANT COPY) What are the linear factors of \(P(x) ?\)

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

RELATING CONCEPTS For individual or group investigation (Exercises \(55-58\) ) The concepts of stretching, shrinking, translating, and reflecting graphs presented earlier 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 downwand 6 units, so the graph should resemble the graph to the right. If we expand the expression \(-2(x+4)^{4}-6\) algebraically, we obtain $$ -2 x^{4}-32 x^{3}-192 x^{2}-512 x-518 $$ 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 finction are given. Sketch by hand the general shape of the graph of the function and describe the transformations. Then support your answer by graphing it on your calculator in a suitable window. $$\begin{array}{l} y=0.5(x-1)^{5}+13 \\ y=0.5 x^{5}-2.5 x^{4}+5 x^{3}-5 x^{2}+2.5 x+12.5 \end{array}$$

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 \(P(x)\). $$P(x)=15 x^{3}+61 x^{2}+2 x-8$$

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 \(P\), given by \(Y_{1}=X^{3}-2 X^{2}-11 X+12.\) (GRAPH CANT COPY) If \(P(x)\) is divided by \(x-2,\) what is the remainder? What is \(P(2) ?\)

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

For the functions in Exercises \(59-66,\) 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. $$P(x)=-2 x^{3}-14 x^{2}+2 x+84$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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