Chapter 3

Solve each equation and inequality. (a) \(4 x\left(x^{2}+1\right)\left(x^{2}+4\right)^{3}+6 x\left(x^{2}+1\right)^{2}\left(x^{2}+4\right)^{2}=0\) (b) \(4 x\left(x^{2}+1\right)\left(x^{2}+4\right)^{3}+6 x\left(x^{2}+1\right)^{2}\left(x^{2}+4\right)^{2}<0\)

Use synthetic division to determine whether the given number is a zero of the polynomial. $$\text { 4; } P(x)=2 x^{3}-6 x^{2}-9 x+6$$

Add or subtract as indicated. Write each sum or difference in standard form. $$(-2+3 i)-(-4+3 i)$$

Solve each equation and inequality, where \(k\) is a positive constant. (a) \(3 k x^{2}-7 x=0\) (b) \(3 k x^{2}-7 x<0\)

Solve each equation. For equations with real solutions, support your answers graphically. $$\frac{1}{3} x^{2}+\frac{1}{4} x-3=0$$

Use the concepts of this section. Show analytically that \(-1\) is a zero of multiplicity 3 of \(P(x)=x^{5}+9 x^{4}+33 x^{3}+55 x^{2}+42 x+12,\) and find all complex zeros. Then, write \(P(x)\) in factored form.

Use synthetic division to determine whether the given number is a zero of the polynomial. $$-4 ; \quad P(x)=9 x^{3}+39 x^{2}+12 x$$

Add or subtract as indicated. Write each sum or difference in standard form. $$(-3+5 i)-(-4+5 i)$$

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

Solve each equation. For equations with real solutions, support your answers graphically. $$\frac{2}{3} x^{2}+\frac{1}{4} x=3$$

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

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

Add or subtract as indicated. Write each sum or difference in standard form. $$(3-8 i)+(2 i+4)$$

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

Solve each equation. For equations with real solutions, support your answers graphically. $$(3-x)^{2}=25$$

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

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

Add or subtract as indicated. Write each sum or difference in standard form. $$(9-5 i)-(3 i-6)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$(2+x)^{2}=49$$

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

Add or subtract as indicated. Write each sum or difference in standard form. $$(2-5 i)-(3+4 i)-(-2+i)$$

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

Solve each equation. For equations with real solutions, support your answers graphically. $$2 x^{2}-4 x=1$$

Use the concepts of this section. Suppose that \(k, a, b,\) and \(c\) are real numbers, \(a \neq 0,\) and a polynomial function \(P(x)\) may be expressed in factored form as \((x-k)\left(a x^{2}+b x+c\right)\). (a) What is the degree of \(P ?\) (b) What are the possible numbers of distinct real zeros of \(P ?\) (c) What are the possible numbers of nonreal complex zeros of \(P ?\) (d) Use the discriminant to explain how to determine the number and type of zeros of \(P\).

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

Add or subtract as indicated. Write each sum or difference in standard form. $$(-4-i)-(2+3 i)+(-4+5 i)$$

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

Solve each equation. For equations with real solutions, support your answers graphically. $$3 x^{2}-6 x=4$$

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

Add or subtract as indicated. Write each sum or difference in standard form. $$(-6+5 i)+(4-4 i)+(2-i)$$

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

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=-1-x$$

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 \(P(x)\). $$P(x)=x^{3}+5 x^{2}+2 x-8$$

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

Add or subtract as indicated. Write each sum or difference in standard form. $$(7+9 i)+(1-2 i)+(-8-7 i)$$

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

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

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 linear factors of \(P(x) ?\)

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

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

Solve each equation. For equations with real solutions, support your answers graphically. $$4 x^{2}-20 x+25=0$$

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 \((-1,-4) ;\) through \((5,104)\)

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 solutions of the equation \(P(x)=0 ?\)

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

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

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

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. $$\begin{array}{l} y=-3(x+1)^{4}+12 \\ y=-3 x^{4}-12 x^{3}-18 x^{2}-12 x+9 \end{array}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$9 x^{2}+12 x+4=0$$