Chapter 3

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. $$x^{3}-3 x^{2}-4 x+12=0 ; \quad[-4,4] \text { by }[-15,15]$$

Explain how you can tell (without graphing it) that the function $$ r(x)=\frac{x^{6}+10}{x^{4}+8 x^{2}+15} $$ has no \(x\) -intercept and no horizontal, vertical, or slant asymptote. What is its end behavior?"

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. $$x^{4}-5 x^{2}+4=0 ; \quad[-4,4] \text { by }[-30,30]$$

In this chapter we adopted the convention that in rational functions, the numerator and denominator don't share a common factor. In this exercise we consider the graph of a rational function that does not satisfy this rule. (a) Show that the graph of $$ r(x)=\frac{3 x^{2}-3 x-6}{x-2} $$ is the line \(y=3 x+3\) with the point \((2,9)\) removed. [Hint: Factor. What is the domain of \(r ?]\) (b) Graph the rational functions: $$ \begin{aligned} s(x) &=\frac{x^{2}+x-20}{x+5} \\ t(x) &=\frac{2 x^{2}-x-1}{x-1} \\ u(x) &=\frac{x-2}{x^{2}-2 x} \end{aligned} $$

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. $$2 x^{4}-5 x^{3}-14 x^{2}+5 x+12=0 ; \quad[-2,5] \text { by }[-40,40]$$

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. $$3 x^{3}+8 x^{2}+5 x+2=0 ; \quad[-3,3] \text { by }[-10,10]$$

Use a graphing device to find all real solutions of the equation, rounded to two decimal places. $$x^{4}-x-4=0$$

Use a graphing device to find all real solutions of the equation, rounded to two decimal places. $$2 x^{3}-8 x^{2}+9 x-9=0$$

Use a graphing device to find all real solutions of the equation, rounded to two decimal places. $$4.00 x^{4}+4.00 x^{3}-10.96 x^{2}-5.88 x+9.09=0$$

Use a graphing device to find all real solutions of the equation, rounded to two decimal places. $$x^{5}+2.00 x^{4}+0.96 x^{3}+5.00 x^{2}+10.00 x+4.80=0$$

Show that the equation $$x^{5}-x^{4}-x^{3}-5 x^{2}-12 x-6=0$$ has exactly one rational root, and then prove that it must have either two or four irrational roots.

Snow began falling at noon on Sunday. The amount of snow on the ground at a certain location at time \(t\) was given by the function $$ \begin{aligned} h(t)=11.60 t &-12.41 t^{2}+6.20 t^{3} \\ &-1.58 t^{4}+0.20 t^{5}-0.01 t^{6}\end{aligned}$$ where \(t\) is measured in days from the start of the snowfall and \(h(t)\) is the depth of snow in inches. Draw a graph of this function, and use your graph to answer the following questions. (a) What happened shortly after noon on Tuesday? (b) Was there ever more than 5 in. of snow on the ground? If so, on what day(s)? (c) On what day and at what time (to the nearest hour) did the snow disappear completely?

Give examples of polynomials that have the following properties, or explain why it is impossible to find such a polynomial. (a) A polynomial of degree 3 that has no real zeros (b) A polynomial of degree 4 that has no real zeros (c) A polynomial of degree 3 that has three real zeros, only one of which is rational (d) A polynomial of degree 4 that has four real zeros, none of which is rational What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?

The most general cubic (thirddegree) equation with rational coefficients can be written as $$x^{3}+a x^{2}+b x+c=0$$ (a) Show that if we replace \(x\) by \(X-a / 3\) and simplify, we end up with an equation that doesn't have an \(X^{2}\) term, that is, an equation of the form $$X^{3}+p X+q=0$$ This is called a depressed cubic, because we have "depressed" the quadratic term. (b) Use the procedure described in part (a) to depress the equation \(x^{3}+6 x^{2}+9 x+4=0\)

The quadratic formula can be used to solve any quadratic (or second-degree) equation. You might have wondered whether similar formulas exist for cubic (thirddegree), quartic (fourth-degree), and higher-degree equations. For the depressed cubic \(x^{3}+p x+q=0,\) Cardano (page 274) found the following formula for one solution: $$x=\sqrt[3]{\frac{-q}{2}+\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}+\sqrt[3]{\frac{-q}{2}-\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}$$ A formula for quartic equations was discovered by the Italian mathematician Ferrari in \(1540 .\) In 1824 the Norwegian mathematician Niels Henrik Abel proved that it is impossible to write a quintic formula, that is, a formula for fifth-degree equations. Finally, Galois (page 254) gave a criterion for determining which equations can be solved by a formula involving radicals. Use the cubic formula to find a solution for the following equations. Then solve the equations using the methods you learned in this section. Which method is easier? (a) \(x^{3}-3 x+2=0\) (b) \(x^{3}-27 x-54=0\) (c) \(x^{3}+3 x+4=0\)