Chapter 3

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$g(x)=\frac{2-x}{x+3}$$

Find an expression for a polynomial \(p(x)\) with real coefficients that satisfies the given conditions. There may be more than one possible answer. Degree \(2 ; x=\frac{1}{2}\) and \(x=\frac{3}{4}\) are zeros

Sketch the polynomial function using transformations. $$f(x)=3 x^{4}+2$$

Find an expression for a polynomial function \(f(x)\) having the given properties. There can be more than one correct anstoer. Degree \(3 ;\) zeros \(-6,0,\) and \(3,\) each of multiplicity 1

Solve the rational inequality. $$\frac{1}{2 x+1} \leq 0$$

Find all real solutions of the polynomial equation. $$x^{4}+x^{3}-x=1$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$g(x)=\frac{x+5}{x-2}$$

Find an expression for a polynomial \(p(x)\) with real coefficients that satisfies the given conditions. There may be more than one possible answer. Degree \(3 ; x=1\) is a zero of multiplicity \(2 ;\) the origin is the \(y\) -intercept

Find the remainder when \(x^{7}+7\) is divided by \(x-1.\)

Sketch the polynomial function using transformations. $$f(x)=-(x+1)^{3}-2$$

Find an expression for a polynomial function \(f(x)\) having the given properties. There can be more than one correct anstoer. Degree \(4 ;\) zeros 2 and \(4,\) cach of multiplicity 2

Solve the rational inequality. $$\frac{-1}{3 x-1}>0$$

Find all real solutions of the polynomial equation. $$6 x^{4}+11 x^{3}-3 x^{2}=2 x$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$g(x)=\frac{x+5}{x-2}$$

Find an expression for a polynomial \(p(x)\) with real coefficients that satisfies the given conditions. There may be more than one possible answer. Degree \(3 ; x=-2\) is a zero of multiplicity \(2 ;\) the origin is an \(x\) -intercept

Find the remainder when \(x^{3}-3\) is divided by \(x+1.\)

Sketch the polynomial function using transformations. $$f(x)=(x-2)^{4}+1$$

Find an expression for a polynomial function \(f(x)\) having the given properties. There can be more than one correct anstoer. Degree \(4 ;\) zeros 2 and \(-3,\) each of multiplicity \(1 ;\) zero at 5 of multiplicity 2

Solve the rational inequality. $$\frac{x+1}{x^{2}-9}<0$$

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=4 x^{4}-5 x^{3}+6 x-3$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$h(x)=\frac{-2 x}{(x-1)(x+4)}$$

Find an expression for a polynomial \(p(x)\) with real coefficients that satisfies the given conditions. There may be more than one possible answer. Degree \(4 ; x=1\) and \(x=\frac{1}{3}\) are both zeros of multiplicity 2

Let \(x-\frac{1}{2}\) be a factor of a polynomial function \(p(x) .\) Find \(p\left(\frac{1}{2}\right).\)

Sketch the polynomial function using transformations. $$h(x)=-\frac{1}{2}(x+1)^{3}-2$$

Find an expression for a polynomial function \(f(x)\) having the given properties. There can be more than one correct anstoer. Degree \(3 ;\) zero at 2 of multiplicity \(1 ;\) zero at -3 of multiplicity 2

Solve the rational inequality. $$\frac{x^{2}-4}{x+5} \geq 0$$

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=x^{4}+6 x^{3}-7 x^{2}+2 x-1$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{x}{(x-3)(x-1)}$$

Find an expression for a polynomial \(p(x)\) with real coefficients that satisfies the given conditions. There may be more than one possible answer. Degree \(4 ; x=-1\) and \(x=-3\) are zeros of multiplicity 1 and \(x=\frac{1}{3}\) is a zero of multiplicity 2

For what value(s) of \(k\) do you get a remainder of 15 when you divide \(k x^{3}+2 x^{2}-10 x+3\) by \(x+2 ?\)

Sketch the polynomial function using transformations. $$h(x)=\frac{1}{2}(x-2)^{4}-1$$

Find an expression for a polynomial function \(f(x)\) having the given properties. There can be more than one correct anstoer. Degree \(3 ;\) zero at 5 of multiplicity 3

Solve the rational inequality. $$\frac{x+1}{x-3} \leq \frac{x-2}{x+4}$$

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=-2 x^{3}+x^{2}-x+1$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{3 x^{2}}{x^{2}-x-2}$$

This set of exercises will draw on the ideas presented in this section and your general math background. One of the zeros of a certain quadratic polynomial with real coefficients is \(1+i .\) What is its other zero?

For what value(s) of \(k\) do you get a remainder of -2 when you divide \(x^{3}-x^{2}+k x+3\) by \(x+1 ?\)

Find an expression for a polynomial function \(f(x)\) having the given properties. There can be more than one correct anstoer. Degree 5 ; zeros at -2 and \(-1,\) each of multiplicity 1 zero at 5 of multiplicity 3

Solve the rational inequality. $$\frac{x-2}{x+2}>\frac{x+5}{x-1}$$

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=-3 x^{3}+2 x^{2}-x-1$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{-4 x^{2}}{x^{2}-x-6}$$

This set of exercises will draw on the ideas presented in this section and your general math background. The graph of a certain cubic polynomial function \(f\) has one \(x\) -intercept at (1,0) that crosses the \(x\) -axis, and another \(x\) -intercept at (-3,0) that touches the \(x\) -axis but does not cross it. What are the zeros of \(f\) and their multiplicities?

Why is the Factor Theorem a direct result of the Remainder Theorem?

Find an expression for a polynomial function \(f(x)\) having the given properties. There can be more than one correct anstoer. Degree 5 ; zeros at -3 and 1 , each of multiplicity \(2 ;\) zero at 4 of multiplicity 1

A rectangular solid has a square base and a height that is 2 inches less than the length of one side of the base. What lengths of the base will produce a volume greater than or equal to 32 inches?

Use Descartes' Rule of Signs to determine the number of positive and negative zeros of \(p\). You need not find the zeros. $$p(x)=2 x^{4}-x^{3}-x^{2}+2 x+5$$

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$f(x)=\frac{x-1}{2 x^{2}-5 x-3}$$

This set of exercises will draw on the ideas presented in this section and your general math background. Explain why there cannot be two different points at which the graph of a cubic polynomial touches the \(x\) -axis without crossing it.

Graph the polynomial function using a graphing utility. Then (a) approximate the \(x\) -intercept(s) of the graph of the function; (b) find the intervals on which the function is positive or negative; (c) approximate the values of \(x\) at which a local maximum or local minimum occurs; and (d) discuss any symmetries. $$f(x)=-x^{3}+3 x+1$$

A rectangular box with a rectangular base is to be built. The length of one side of the rectangular base is 3 inches more than the height of the box, while the length of the other side of the rectangular base is 1 inch more than the height. For what values of the height will the volume of the box be greater than or equal to 40 cubic inches?