Chapter 9

For Problems \(35-42\), (a) find the \(y\) intercepts, (b) find the \(x\) intercepts, and (c) find the intervals of \(x\) where \(f(x)>0\) and those where \(f(x)<0\). Do not sketch the graphs. $$ f(x)=x(x-6)^{2}(x+4) $$

For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x+1, \quad f(x)=x^{3}-2 x^{2}-7 x-4 $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(9 x^{3}-6 x^{2}+3 x-4\right) \div\left(x-\frac{1}{3}\right) $$

For Problems \(35-42\), (a) find the \(y\) intercepts, (b) find the \(x\) intercepts, and (c) find the intervals of \(x\) where \(f(x)>0\) and those where \(f(x)<0\). Do not sketch the graphs. $$ f(x)=(x+2)^{2}(x-1)^{3}(x-2) $$

For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x-5, \quad f(x)=2 x^{3}+x^{2}-61 x+30 $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(2 x^{3}+3 x^{2}-2 x+3\right) \div\left(x+\frac{1}{2}\right) $$

For Problems \(35-42\), (a) find the \(y\) intercepts, (b) find the \(x\) intercepts, and (c) find the intervals of \(x\) where \(f(x)>0\) and those where \(f(x)<0\). Do not sketch the graphs. $$ f(x)=x^{2}(2-x)(x+3) $$

Explain what it means to say that the equation \((x+3)^{2}=\) 0 has a solution of \(-3\) with a multiplicity of two.

For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x-6, \quad f(x)=x^{5}-6 x^{4}-16 x+96 $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(3 x^{4}-2 x^{3}+5 x^{2}-x-1\right) \div\left(x+\frac{1}{3}\right) $$

For Problems \(35-42\), (a) find the \(y\) intercepts, (b) find the \(x\) intercepts, and (c) find the intervals of \(x\) where \(f(x)>0\) and those where \(f(x)<0\). Do not sketch the graphs. $$ f(x)=(x+2)^{5}(x-4)^{2} $$

Describe how to use the rational root theorem to show that the equation \(x^{2}-3=0\) has no rational solutions.

For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x+3, \quad f(x)=x^{5}+3 x^{4}-x-3 $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(4 x^{4}-5 x^{2}+1\right) \div\left(x-\frac{1}{2}\right) $$

Use the rational root theorem to argue that \(\sqrt{2}\) is not a rational number. [Hint: The solutions of \(x^{2}-2=0\) are \(\pm \sqrt{2}\).]

For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x+5, \quad f(x)=9 x^{3}+21 x^{2}-104 x+80 $$

How would you give a general description of what is accomplished with synthetic division to someone who had just completed an elementary algebra course?

Use the rational root theorem to argue that \(\sqrt{12}\) is not a rational number.

For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x+4, \quad f(x)=4 x^{3}+4 x^{2}-39 x+36 $$

Why is synthetic division restricted to situations where the divisor is of the form \(x-c\) ?

A polynomial function with real coefficients is continuous everywhere; that is, its graph has no holes or breaks. This is the basis for the following property: If \(f(x)\) is a polynomial with real coefficients, and if \(f(a)\) and \(f(b)\) are of opposite sign, then there is at least one real zero between \(a\) and \(b\). This property, along with our knowledge of polynomial functions, provides the basis for locating and approximating irrational solutions of a polynomial equation. Consider the equation \(x^{3}+2 x-4=0\). Applying Descartes' rule of signs, we can determine that this equation has one positive real solution and two nonreal complex solutions. (You may want to confirm this!) The rational root theorem indicates that the only possible rational solutions are 1,2 , and 4 . Using a little more compact format for synthetic division, we obtain the following results when testing for 1 and 2 as possible solutions: $$ \begin{array}{r|rrrr} & 1 & 0 & 2 & -4 \\ 1 & 1 & 1 & 3 & -1 \\ 2 & 1 & 2 & 6 & 8 \end{array} $$ Because \(f(1)=-1\) and \(f(2)=8\), there must be an irrational solution between 1 and 2 . Furthermore, \(-1\) is closer to 0 than is 8 , so our guess is that the solution is closer to 1 than to 2 . Let's start looking at \(1.0,1.1,1.2\), and so on, until we can place the solution between two numbers. Because \(f(1.1)=-0.469\) and \(f(1.2)=0.128\), the irrational solution must be between \(1.1\) and 1.2. Furthermore, because \(0.128\) is closer to 0 than is \(-0.469\), our guess is that the solution is closer to \(1.2\) than to \(1.1\). Let's start looking at \(1.15,1.16\), and so on. $$ \begin{array}{l|rrrrr} & 1 & 0 & 2 & -4 \\ \ 1.15 & 1 & 1.15 & 3.3225 & -0.179 \\ 1.16 & 1 & 1.16 & 3.3456 & -0.119 \\ 1.17 & 1 & 1.17 & 3.3689 & -0.058 \\ 1.18 & 1 & 1.18 & 3.3924 & 0.003 \end{array} $$ Because \(f(1.17)=-0.058\) and \(f(1.18)=0.003\), the irrational solution must be between \(1.17\) and \(1.18\). Therefore we can use \(1.2\) as a rational approximation to the nearest tenth. For each of the following equations, (a) verify that the equation has exactly one irrational solution, and (b) find an approximation, to the nearest tenth, of that solution. (a) \(x^{3}+x-6=0 \) (b) \(x^{3}-6 x-6=0 \) (c) \(x^{3}-27 x-60=0 \) (d) \(x^{3}-x^{2}-x-1=0 \) (e) \(x^{3}-2 x-10=0\) (f) \(x^{3}-5 x^{2}-1=0 \)

Defend this statement: "Every polynomial equation of odd degree with real coefficients has at least one real number solution."

Graph \(f(x)=x^{3}\). Now predict the graphs for \(f(x)=\) \(x^{3}+2, f(x)=-x^{3}+2\), and \(f(x)=-x^{3}-2\). Graph these three functions on the same set of axes with the graph of \(f(x)=x^{3}\).

For Problems \(45-48\), find the value(s) of \(k\) that makes the second polynomial a factor of the first. $$ x^{3}-k x^{2}+5 x+k ; x-2 $$

Draw a rough sketch of the graphs of the functions \(f(x)=x^{3}-x^{2}, f(x)=-x^{3}+x^{2}\), and \(f(x)=-x^{3}-x^{2} .\) Now graph these three functions to check your sketches.

Solve each of the following equations, using a graphing calculator whenever it seems to be helpful. Express all irrational solutions in lowest radical form. (a) \(x^{3}+2 x^{2}-14 x-40=0\) (b) \(x^{3}+x^{2}-7 x+65=0\) (c) \(x^{4}-6 x^{3}-6 x^{2}+32 x+24=0\) (d) \(x^{4}+3 x^{3}-39 x^{2}+11 x+24=0\) (e) \(x^{3}-14 x^{2}+26 x-24=0\) (f) \(x^{4}+2 x^{3}-3 x^{2}-4 x+4=0\)

For Problems \(45-48\), find the value(s) of \(k\) that makes the second polynomial a factor of the first. $$ k x^{3}+19 x^{2}+x-6 ; x+3 $$

Graph \(f(x)=x^{4}+x^{3}+x^{2}\). What should the graphs of \(f(x)=x^{4}-x^{3}+x^{2}\) and \(f(x)=-x^{4}-x^{3}-x^{2}\) look like? Graph them to see if you were right.

Find approximations, to the nearest hundredth, of the real number solutions of each of the following equations: (a) \(x^{2}-4 x+1=0\) (b) \(3 x^{3}-2 x^{2}+12 x-8=0\) (c) \(x^{4}-8 x^{3}+14 x^{2}-8 x+13=0\) (d) \(x^{4}+6 x^{3}-10 x^{2}-22 x+161=0\) (e) \(7 x^{5}-5 x^{4}+35 x^{3}-25 x^{2}+28 x-20=0\)

For Problems \(45-48\), find the value(s) of \(k\) that makes the second polynomial a factor of the first. $$ x^{3}+4 x^{2}-11 x+k ; x+2 $$

How should the graphs of \(f(x)=x^{3}, f(x)=x^{5}\), and \(f(x)=x^{7}\) compare? Graph these three functions on the same set of axes.

Argue that \(f(x)=3 x^{4}+2 x^{2}+5\) has no factor of the form \(x-c\), where \(c\) is a real number.

How should the graphs of \(f(x)=x^{2}, f(x)=x^{4}\), and \(f(x)=x^{6}\) compare? Graph these three functions on the same set of axes.

Show that \(x+2\) is a factor of \(x^{12}-4096\).

For each of the following functions, find the \(x\) intercepts, and find the intervals of \(x\) where \(f(x)>0\) and those where \(f(x)<0\). (a) \(f(x)=x^{3}-3 x^{2}-6 x+8\) (b) \(f(x)=x^{3}-8 x^{2}-x+8\) (c) \(f(x)=x^{3}-7 x^{2}+16 x-12\) (d) \(f(x)=x^{3}-19 x^{2}+90 x-72\) (e) \(f(x)=x^{4}+3 x^{3}-3 x^{2}-11 x-6\) (f) \(f(x)=x^{4}+12 x^{2}-64\)

Verify that \(x+1\) is a factor of \(x^{n}-1\) for all even positive integral values of \(n\).

Find the coordinates of the turning points of each of the following graphs. Express \(x\) and \(y\) values to the nearest integer. (a) \(f(x)=2 x^{3}-3 x^{2}-12 x+40\) (b) \(f(x)=2 x^{3}-33 x^{2}+60 x+1050\) (c) \(f(x)=-2 x^{3}-9 x^{2}+24 x+100\) (d) \(f(x)=x^{4}-4 x^{3}-2 x^{2}+12 x+3\) (e) \(f(x)=x^{3}-30 x^{2}+288 x-900\) (f) \(f(x)=x^{5}-2 x^{4}-3 x^{3}-2 x^{2}+x-1\)

Verify that \(x-1\) is a factor of \(x^{n}-1\) for all positive integral values of \(n\). See below

For each of the following functions, find the \(x\) intercepts and find the turning points. Express your answers to the nearest tenth. (a) \(f(x)=x^{3}+2 x^{2}-3 x+4\) (b) \(f(x)=42 x^{3}-x^{2}-246 x-35\) (c) \(f(x)=x^{4}-4 x^{2}-4\)

(a) Verify that \(x-y\) is a factor of \(x^{n}-y^{n}\) for all positive integral values of \(n\). See below (b) Verify that \(x+y\) is a factor of \(x^{n}-y^{n}\) for all even positive integral values of \(n\). See below (c) Verify that \(x+y\) is a factor of \(x^{n}+y^{n}\) for all odd positive integral values of \(n\). See below

A rectangular piece of cardboard is 13 inches long and 9 inches wide. From each corner, a square piece is cut out, and then the flaps are turned up to form an open box. Determine the length of a side of the square pieces so that the volume of the box is as large as possible.

State the remainder theorem in your own words.

A company determines that its weekly profit from manufacturing and selling \(x\) units of a certain item is given by \(P(x)=-x^{3}+3 x^{2}+2880 x-500\). What weekly production rate will maximize the profit?

Discuss some of the uses of the factor theorem.

The remainder and factor theorems are true for any complex value of \(c\). Therefore, for Problems \(56-58\), find \(f(c)\) by (a) using synthetic division and the remainder theorem, and (b) evaluating \(f(c)\) directly. $$ f(x)=x^{3}-5 x^{2}+2 x+1 \text { and } c=i $$

The remainder and factor theorems are true for any complex value of \(c\). Therefore, for Problems \(56-58\), find \(f(c)\) by (a) using synthetic division and the remainder theorem, and (b) evaluating \(f(c)\) directly. $$ f(x)=x^{2}+4 x-2 \text { and } c=1+i $$

The remainder and factor theorems are true for any complex value of \(c\). Therefore, for Problems \(56-58\), find \(f(c)\) by (a) using synthetic division and the remainder theorem, and (b) evaluating \(f(c)\) directly. $$ f(x)=x^{3}+2 x^{2}+x-2 \text { and } c=2-3 i $$

The remainder and factor theorems are true for any complex value of \(c\). Therefore, for Problems \(56-58\), find \(f(c)\) by (a) using synthetic division and the remainder theorem, and (b) evaluating \(f(c)\) directly. $$ \text { Show that } x+3 i \text { is a factor of } f(x)=x^{4}+14 x^{2}+45 \text {. } $$