Algebra and Trigonometry

# Review Exercises

Algebra and TrigonometryReview Exercises

#### Real Numbers: Algebra Essentials

For the following exercises, perform the given operations.

1.

$( 5−3⋅2 ) 2 −6 ( 5−3⋅2 ) 2 −6$

2.

$64÷( 2⋅8 )+14÷7 64÷( 2⋅8 )+14÷7$

3.

$2⋅ 5 2 +6÷2 2⋅ 5 2 +6÷2$

For the following exercises, solve the equation.

4.

$5x+9=−11 5x+9=−11$

5.

$2y+ 4 2 =64 2y+ 4 2 =64$

For the following exercises, simplify the expression.

6.

$9( y+2 )÷3⋅2+1 9( y+2 )÷3⋅2+1$

7.

$3m( 4+7 )−m 3m( 4+7 )−m$

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

8.

11

9.

0

10.

$5 6 5 6$

11.

$11 11$

#### Exponents and Scientific Notation

For the following exercises, simplify the expression.

12.

$2 2 ⋅ 2 4 2 2 ⋅ 2 4$

13.

$4 5 4 3 4 5 4 3$

14.

$( a 2 b 3 ) 4 ( a 2 b 3 ) 4$

15.

$6 a 2 ⋅ a 0 2 a −4 6 a 2 ⋅ a 0 2 a −4$

16.

$( xy ) 4 y 3 ⋅ 2 x 5 ( xy ) 4 y 3 ⋅ 2 x 5$

17.

$4 −2 x 3 y −3 2 x 0 4 −2 x 3 y −3 2 x 0$

18.

$( 2 x 2 y ) −2 ( 2 x 2 y ) −2$

19.

$( 16 a 3 b 2 ) ( 4a b −1 ) −2 ( 16 a 3 b 2 ) ( 4a b −1 ) −2$

20.

Write the number in standard notation:$2.1314 × 10 −6 2.1314 × 10 −6$

21.

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

For the following exercises, find the principal square root.

22.

$121 121$

23.

$196 196$

24.

$361 361$

25.

$75 75$

26.

$162 162$

27.

$32 25 32 25$

28.

$80 81 80 81$

29.

$49 1250 49 1250$

30.

$2 4+ 2 2 4+ 2$

31.

$4 3 +6 3 4 3 +6 3$

32.

$12 5 −13 5 12 5 −13 5$

33.

$−243 5 −243 5$

34.

$250 3 −8 3 250 3 −8 3$

#### Polynomials

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

35.

$(3 x 3 +2x−1)+(4 x 2 −2x+7) (3 x 3 +2x−1)+(4 x 2 −2x+7)$

36.

$( 2y+1 )−( 2 y 2 −2y−5 ) ( 2y+1 )−( 2 y 2 −2y−5 )$

37.

$(2 x 2 +3x−6)+(3 x 2 −4x+9) (2 x 2 +3x−6)+(3 x 2 −4x+9)$

38.

$( 6 a 2 +3a+10 )−( 6 a 2 −3a+5 ) ( 6 a 2 +3a+10 )−( 6 a 2 −3a+5 )$

39.

$(k+3)(k−6) (k+3)(k−6)$

40.

$(2h+1)(3h−2) (2h+1)(3h−2)$

41.

$( x+1 )( x 2 +1 ) ( x+1 )( x 2 +1 )$

42.

$(m−2)( m 2 +2m−3) (m−2)( m 2 +2m−3)$

43.

$( a+2b )( 3a−b ) ( a+2b )( 3a−b )$

44.

$( x+y )( x−y ) ( x+y )( x−y )$

#### Factoring Polynomials

For the following exercises, find the greatest common factor.

45.

$81p+9pq−27 p 2 q 2 81p+9pq−27 p 2 q 2$

46.

$12 x 2 y+4x y 2 −18xy 12 x 2 y+4x y 2 −18xy$

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.

$2 x 2 −9x−18 2 x 2 −9x−18$

49.

$8 a 2 +30a−27 8 a 2 +30a−27$

50.

$d 2 −5d−66 d 2 −5d−66$

51.

$x 2 +10x+25 x 2 +10x+25$

52.

$y 2 −6y+9 y 2 −6y+9$

53.

$4 h 2 −12hk+9 k 2 4 h 2 −12hk+9 k 2$

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.

$4x (x−1) − 1 4 +3 (x−1) 3 4 4x (x−1) − 1 4 +3 (x−1) 3 4$

59.

$3p ( p+3 ) 1 3 −8 ( p+3 ) 4 3 3p ( p+3 ) 1 3 −8 ( p+3 ) 4 3$

60.

$4r ( 2r−1 ) − 2 3 −5 ( 2r−1 ) 1 3 4r ( 2r−1 ) − 2 3 −5 ( 2r−1 ) 1 3$

#### Rational Expressions

For the following exercises, simplify the expression.

61.

$x 2 −x−12 x 2 −8x+16 x 2 −x−12 x 2 −8x+16$

62.

$4 y 2 −25 4 y 2 −20y+25 4 y 2 −25 4 y 2 −20y+25$

63.

$2 a 2 −a−3 2 a 2 −6a−8 ⋅ 5 a 2 −19a−4 10 a 2 −13a−3 2 a 2 −a−3 2 a 2 −6a−8 ⋅ 5 a 2 −19a−4 10 a 2 −13a−3$

64.

$d−4 d 2 −9 ⋅ d−3 d 2 −16 d−4 d 2 −9 ⋅ d−3 d 2 −16$

65.

$m 2 +5m+6 2 m 2 −5m−3 ÷ 2 m 2 +3m−9 4 m 2 −4m−3 m 2 +5m+6 2 m 2 −5m−3 ÷ 2 m 2 +3m−9 4 m 2 −4m−3$

66.

$4 d 2 −7d−2 6 d 2 −17d+10 ÷ 8 d 2 +6d+1 6 d 2 +7d−10 4 d 2 −7d−2 6 d 2 −17d+10 ÷ 8 d 2 +6d+1 6 d 2 +7d−10$

67.

$10 x + 6 y 10 x + 6 y$

68.

$12 a 2 +2a+1 − 3 a 2 −1 12 a 2 +2a+1 − 3 a 2 −1$

69.

$1 d + 2 c 6c+12d dc 1 d + 2 c 6c+12d dc$

70.

$3 x − 7 y 2 x 3 x − 7 y 2 x$