Jay Abramson

Review Exercises

You have reached the end of Chapter 3: Polynomial and Rational Functions. Let’s review some of the Key Terms, Concepts and Equations you have learned.

Complex Numbers

Perform the indicated operation with complex numbers.

$(4 + 3 i) + (- 2 - 5 i)$

$(6 - 5 i) - (10 + 3 i)$

$(2 - 3 i) (3 + 6 i)$

$\frac{2 - i}{2 + i}$

Solve the following equations over the complex number system.

$x^{2} - 4 x + 5 = 0$

$x^{2} + 2 x + 10 = 0$

Quadratic Functions

For the following exercises, write the quadratic function in standard form. Then, give the vertex and axes intercepts. Finally, graph the function.

$f (x) = x^{2} - 4 x - 5$

$f (x) = - 2 x^{2} - 4 x$

For the following problems, find the equation of the quadratic function using the given information.

The vertex is $(- 2, 3)$ and a point on the graph is $(3, 6) .$

10.

The vertex is $(- 3, 6.5)$ and a point on the graph is $(2, 6) .$

Answer the following questions.

11.

A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.

12.

An object projected from the ground at a 45 degree angle with initial velocity of 120 feet per second has height, $h,$ in terms of horizontal distance traveled, $x,$ given by $h (x) = \frac{- 32}{{(120)}^{2}} x^{2} + x .$ Find the maximum height the object attains.

Power Functions and Polynomial Functions

For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.

13.

$f (x) = 4 x^{5} - 3 x^{3} + 2 x - 1$

14.

$f (x) = 5^{x + 1} - x^{2}$

15.

$f (x) = x^{2} (3 - 6 x + x^{2})$

For the following exercises, determine end behavior of the polynomial function.

16.

$f (x) = 2 x^{4} + 3 x^{3} - 5 x^{2} + 7$

17.

$f (x) = 4 x^{3} - 6 x^{2} + 2$

18.

$f (x) = 2 x^{2} (1 + 3 x - x^{2})$

Graphs of Polynomial Functions

For the following exercises, find all zeros of the polynomial function, noting multiplicities.

19.

$f (x) = {(x + 3)}^{2} (2 x - 1) {(x + 1)}^{3}$

20.

$f (x) = x^{5} + 4 x^{4} + 4 x^{3}$

21.

$f (x) = x^{3} - 4 x^{2} + x - 4$

For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.

22.

Graph of an odd-degree polynomial with two turning points.

23.

Graph of an even-degree polynomial with two turning points.

24.

Use the Intermediate Value Theorem to show that at least one zero lies between 2 and 3 for the function $f (x) = x^{3} - 5 x + 1$

Dividing Polynomials

For the following exercises, use long division to find the quotient and remainder.

25.

$\frac{x^{3} - 2 x^{2} + 4 x + 4}{x - 2}$

26.

$\frac{3 x^{4} - 4 x^{2} + 4 x + 8}{x + 1}$

For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

27.

$\frac{x^{3} - 2 x^{2} + 5 x - 1}{x + 3}$

28.

$\frac{x^{3} + 4 x + 10}{x - 3}$

29.

$\frac{2 x^{3} + 6 x^{2} - 11 x - 12}{x + 4}$

30.

$\frac{3 x^{4} + 3 x^{3} + 2 x + 2}{x + 1}$

Zeros of Polynomial Functions

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.

31.

$2 x^{3} - 3 x^{2} - 18 x - 8 = 0$

32.

$3 x^{3} + 11 x^{2} + 8 x - 4 = 0$

33.

$2 x^{4} - 17 x^{3} + 46 x^{2} - 43 x + 12 = 0$

34.

$4 x^{4} + 8 x^{3} + 19 x^{2} + 32 x + 12 = 0$

For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

35.

$x^{3} - 3 x^{2} - 2 x + 4 = 0$

36.

$2 x^{4} - x^{3} + 4 x^{2} - 5 x + 1 = 0$

Rational Functions

For the following rational functions, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph.

37.

$f (x) = \frac{x + 2}{x - 5}$

38.

$f (x) = \frac{x^{2} + 1}{x^{2} - 4}$

39.

$f (x) = \frac{3 x^{2} - 27}{x^{2} + x - 2}$

40.

$f (x) = \frac{x + 2}{x^{2} - 9}$

For the following exercises, find the slant asymptote.

41.

$f (x) = \frac{x^{2} - 1}{x + 2}$

42.

$f (x) = \frac{2 x^{3} - x^{2} + 4}{x^{2} + 1}$

Inverses and Radical Functions

For the following exercises, find the inverse of the function with the domain given.

43.

$f (x) = {(x - 2)}^{2}, x \geq 2$

44.

$f (x) = {(x + 4)}^{2} - 3, x \geq - 4$

45.

$f (x) = x^{2} + 6 x - 2, x \geq - 3$

46.

$f (x) = 2 x^{3} - 3$

47.

$f (x) = \sqrt{4 x + 5} - 3$

48.

$f (x) = \frac{x - 3}{2 x + 1}$

Modeling Using Variation

For the following exercises, find the unknown value.

49.

$y$ varies directly as the square of $x .$ If when $x = 3, y = 36,$ find $y$ if $x = 4.$

50.

$y$ varies inversely as the square root of $x$ If when $x = 25, y = 2,$ find $y$ if $x = 4.$

51.

$y$ varies jointly as the cube of $x$ and as $z .$ If when $x = 1$ and $z = 2,$ $y = 6,$ find $y$ if $x = 2$ and $z = 3.$

52.

$y$ varies jointly as $x$ and the square of $z$ and inversely as the cube of $w .$ If when $x = 3,$ $z = 4,$ and $w = 2,$ $y = 48,$ find $y$ if $x = 4,$ $z = 5,$ and $w = 3.$

For the following exercises, solve the application problem.

53.

The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.

54.

The volume $V$ of an ideal gas varies directly with the temperature $T$ and inversely with the pressure $P$ . A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.