Sumário

Preface
1Prerequisites
2Equations and Inequalities
3Functions
4Linear Functions
5Polynomial and Rational Functions
6Exponential and Logarithmic Functions
7The Unit Circle: Sine and Cosine Functions
8Periodic Functions
9Trigonometric Identities and Equations
10Further Applications of Trigonometry
11Systems of Equations and Inequalities
12Analytic Geometry
13Sequences, Probability, and Counting Theory
AProofs, Identities, and Toolkit Functions
Answer Key
Index

Review Exercises

Real Numbers: Algebra Essentials

For the following exercises, perform the given operations.

${(5 - 3 \cdot 2)}^{2} - 6 {(5 - 3 \cdot 2)}^{2} - 6$

$64 \div (2 \cdot 8) + 14 \div 7 64 \div (2 \cdot 8) + 14 \div 7$

$2 \cdot 5^{2} + 6 \div 2 2 \cdot 5^{2} + 6 \div 2$

For the following exercises, solve the equation.

$5 x + 9 = −11 5 x + 9 = −11$

$2 y + 4^{2} = 64 2 y + 4^{2} = 64$

For the following exercises, simplify the expression.

$9 (y + 2) \div 3 \cdot 2 + 1 9 (y + 2) \div 3 \cdot 2 + 1$

$3 m (4 + 7) - m 3 m (4 + 7) - m$

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

10.

$\frac{5}{6} \frac{5}{6}$

11.

$\sqrt{11} \sqrt{11}$

Exponents and Scientific Notation

For the following exercises, simplify the expression.

12.

$2^{2} \cdot 2^{4} 2^{2} \cdot 2^{4}$

13.

$\frac{4^{5}}{4^{3}} \frac{4^{5}}{4^{3}}$

14.

${(\frac{a^{2}}{b^{3}})}^{4} {(\frac{a^{2}}{b^{3}})}^{4}$

15.

$\frac{6 a^{2} \cdot a^{0}}{2 a^{−4}} \frac{6 a^{2} \cdot a^{0}}{2 a^{−4}}$

16.

$\frac{{(x y)}^{4}}{y^{3}} \cdot \frac{2}{x^{5}} \frac{{(x y)}^{4}}{y^{3}} \cdot \frac{2}{x^{5}}$

17.

$\frac{4^{−2} x^{3} y^{−3}}{2 x^{0}} \frac{4^{−2} x^{3} y^{−3}}{2 x^{0}}$

18.

${(\frac{2 x^{2}}{y})}^{−2} {(\frac{2 x^{2}}{y})}^{−2}$

19.

$(\frac{16 a^{3}}{b^{2}}) {(4 a b^{−1})}^{−2} (\frac{16 a^{3}}{b^{2}}) {(4 a b^{−1})}^{−2}$

20.

Write the number in standard notation: $2.1314 \times 10^{−6} 2.1314$

21.

Write the number in scientific notation: 16,340,000

Radicals and Rational Expressions

For the following exercises, find the principal square root.

22.

$\sqrt{121} \sqrt{121}$

23.

$\sqrt{196} \sqrt{196}$

24.

$\sqrt{361} \sqrt{361}$

25.

$\sqrt{75} \sqrt{75}$

26.

$\sqrt{162} \sqrt{162}$

27.

$\sqrt{\frac{32}{25}} \sqrt{\frac{32}{25}}$

28.

$\sqrt{\frac{80}{81}} \sqrt{\frac{80}{81}}$

29.

$\sqrt{\frac{49}{1250}} \sqrt{\frac{49}{1250}}$

30.

$\frac{2}{4 + \sqrt{2}} \frac{2}{4 + \sqrt{2}}$

31.

$4 \sqrt{3} + 6 \sqrt{3} 4 \sqrt{3} + 6 \sqrt{3}$

32.

$12 \sqrt{5} - 13 \sqrt{5} 12 \sqrt{5} - 13 \sqrt{5}$

33.

$\sqrt[5]{−243} \sqrt[5]{−243}$

34.

$\frac{\sqrt[3]{250}}{\sqrt[3]{−8}} \frac{\sqrt[3]{250}}{\sqrt[3]{−8}}$

Polynomials

For the following exercises, perform the given operations and simplify.

35.

$(3 x^{3} + 2 x - 1) + (4 x^{2} - 2 x + 7) (3 x^{3} + 2 x - 1) + (4 x^{2} - 2 x + 7)$

36.

$(2 y + 1) - (2 y^{2} - 2 y - 5) (2 y + 1) - (2 y^{2} - 2 y - 5)$

37.

$(2 x^{2} + 3 x - 6) + (3 x^{2} - 4 x + 9) (2 x^{2} + 3 x - 6) + (3 x^{2} - 4 x + 9)$

38.

$(6 a^{2} + 3 a + 10) - (6 a^{2} −3 a + 5) (6 a^{2} + 3 a + 10) - (6 a^{2} −3 a + 5)$

39.

$(k + 3) (k - 6) (k + 3) (k - 6)$

40.

$(2 h + 1) (3 h - 2) (2 h + 1) (3 h - 2)$

41.

$(x + 1) (x^{2} + 1) (x + 1) (x^{2} + 1)$

42.

$(m - 2) (m^{2} + 2 m - 3) (m - 2) (m^{2} + 2 m - 3)$

43.

$(a + 2 b) (3 a - b) (a + 2 b) (3 a - b)$

44.

$(x + y) (x - y) (x + y) (x - y)$

Factoring Polynomials

For the following exercises, find the greatest common factor.

45.

$81 p + 9 p q - 27 p^{2} q^{2} 81 p + 9 p q - 27 p^{2} q^{2}$

46.

$12 x^{2} y + 4 x y^{2} −18 x y 12 x^{2} y + 4 x y^{2} −18 x y$

47.

$88 a^{3} b + 4 a^{2} b - 144 a^{2} 88 a^{3} b + 4 a^{2} b - 144 a^{2}$

For the following exercises, factor the polynomial.

48.

$2 x^{2} - 9 x - 18 2 x^{2} - 9 x - 18$

49.

$8 a^{2} + 30 a - 27 8 a^{2} + 30 a - 27$

50.

$d^{2} - 5 d - 66 d^{2} - 5 d - 66$

51.

$x^{2} + 10 x + 25 x^{2} + 10 x + 25$

52.

$y^{2} - 6 y + 9 y^{2} - 6 y + 9$

53.

$4 h^{2} - 12 h k + 9 k^{2} 4 h^{2} - 12 h k + 9 k^{2}$

54.

$361 x^{2} - 121 361 x^{2} - 121$

55.

$p^{3} + 216 p^{3} + 216$

56.

$8 x^{3} - 125 8 x^{3} - 125$

57.

$64 q^{3} - 27 p^{3} 64 q^{3} - 27 p^{3}$

58.

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

59.

$3 p {(p + 3)}^{\frac{1}{3}} −8 {(p + 3)}^{\frac{4}{3}} 3 p {(p + 3)}^{\frac{1}{3}} −8 {(p + 3)}^{\frac{4}{3}}$

60.

$4 r {(2 r - 1)}^{- \frac{2}{3}} - 5 {(2 r - 1)}^{\frac{1}{3}} 4 r {(2 r - 1)}^{- \frac{2}{3}} - 5 {(2 r - 1)}^{\frac{1}{3}}$

Rational Expressions

For the following exercises, simplify the expression.

61.

$\frac{x^{2} - x - 12}{x^{2} - 8 x + 16} \frac{x^{2} - x - 12}{x^{2} - 8 x + 16}$

62.

$\frac{4 y^{2} - 25}{4 y^{2} - 20 y + 25} \frac{4 y^{2} - 25}{4 y^{2} - 20 y + 25}$

63.

$\frac{2 a^{2} - a - 3}{2 a^{2} - 6 a - 8} \cdot \frac{5 a^{2} - 19 a - 4}{10 a^{2} - 13 a - 3} \frac{2 a^{2} - a - 3}{2 a^{2} - 6 a - 8} \cdot \frac{5 a^{2} - 19 a - 4}{10 a^{2} - 13 a - 3}$

64.

$\frac{d - 4}{d^{2} - 9} \cdot \frac{d - 3}{d^{2} - 16} \frac{d - 4}{d^{2} - 9} \cdot \frac{d - 3}{d^{2} - 16}$

65.

$\frac{m^{2} + 5 m + 6}{2 m^{2} - 5 m - 3} \div \frac{2 m^{2} + 3 m - 9}{4 m^{2} - 4 m - 3} \frac{m^{2} + 5 m + 6}{2 m^{2} - 5 m - 3} \div \frac{2 m^{2} + 3 m - 9}{4 m^{2} - 4 m - 3}$

66.

$\frac{4 d^{2} - 7 d - 2}{6 d^{2} - 17 d + 10} \div \frac{8 d^{2} + 6 d + 1}{6 d^{2} + 7 d - 10} \frac{4 d^{2} - 7 d - 2}{6 d^{2} - 17 d + 10} \div \frac{8 d^{2} + 6 d + 1}{6 d^{2} + 7 d - 10}$

67.

$\frac{10}{x} + \frac{6}{y} \frac{10}{x} + \frac{6}{y}$

68.

$\frac{12}{a^{2} + 2 a + 1} - \frac{3}{a^{2} −1} \frac{12}{a^{2} + 2 a + 1} - \frac{3}{a^{2} −1}$

69.

$\frac{\frac{1}{d} + \frac{2}{c}}{\frac{6 c + 12 d}{d c}} \frac{\frac{1}{d} + \frac{2}{c}}{\frac{6 c + 12 d}{d c}}$

70.

$\frac{\frac{3}{x} - \frac{7}{y}}{\frac{2}{x}} \frac{\frac{3}{x} - \frac{7}{y}}{\frac{2}{x}}$