Trigonometry Algebra 2 Review - HASD

Algebra 2 Review Packet Name

Trigonometry

Algebra 2 Review

On the following pages you will find a review of the Algebra 2 concepts needed to successfully complete

Trigonometry. Concepts such as fractions, proportions, factoring, solving polynomials, simplifying square

roots, complex numbers, the quadratic formula, rational exponents, and rational expressions are contained in

this packet. You are expected to have full function of these concepts, because they will not be reviewed

throughout the year.

The completed packet will be turned in the first day of school. On the second day of school a test will be

administered to assess basic Algebra 2 concepts. A minimum of 70% is required to remain in Trigonometry.

Students who do not attain a 70% will be removed from the roster and rescheduled.

Although all of you have taken Algebra 2, not all of you have learned or remember the concepts in this packet.

I recommend using your Algebra 2 notebook as a resource to complete the packet. If you do not have your

notebook, then refer to Khan Academy (www.khanacademy.org ) for video instruction.

The packet should take about 3 hours to complete. I recommend waiting until August to complete the packet, in

order to ensure the concepts are easily recalled for the test.

Mrs. Amoriello

[email protected]

http://www.khanacademy.org/

mailto:[email protected]



Fractions No Calculator! Show all your work for each problem.

Add or Subtract.

1. 12

5

8

3 2.

12

1

4

7 3.

2

1

4

3 4.

2

1

8

7

5. 3

1

5

2 6.

6

5

9

8 7.

14

2

7

6 8.

6

1

8

3

Multiply or Divide.

9. 5

4

3

2 10.

15

4

8

5 11.

6

5

4

3 12.

6

58

\

13. 2

1

12

5 14.

10

3

8

7 15.

12

7

16

21 16.

16

15

24

25

Complex Fractions: take the reciprocal of the denominator and multiply. Ex: 10

3

5

2

4

3

2

54

3

17. 3

9

5

18. 7

3

6

19.

15

146

7

20.

10

97

6

21.

23

13

1

22.

9

7

81

7

23.

5

16

36 24.

8

19

32

1


Proportions

Ex: Solve the proportion. x

10

6

5

1065 x Cross Multiply.

605 x

12x

Solve the proportion. Write the answer in simplified fraction form.

1. x

4

2

6 2.

3

84

x 3.

10

7

5

x

4. x5

2

5

6 5.

6

6

8

9

x 6.

3

4

10

2

x

7. 3

4

2

10

x 8.

1

8

3

2

x 9.

9

7

8

xx

10. x

x 10

4

9 11.

4

7

5

9

x

x 12.

5

4

9

6

xx

2


Factoring Special Products

Greatest Common Factor

Factor out the greatest common factor found in each term.

Ex: 93 x Ex: xx 164 2 Ex: 1068 2 xx

333 x 444 xxx 523242 2 xx

33 x 44 xx 5342 2 xx

1. 4035 x 2. 144 x 3. 69 2 x

4. xx 52 5. xx 1812 2 6. 24 246 xx

7. 4164 2 xx 8. 234 24183 xxx 9. 15305 2 xx

Difference of Two Squares

bababa 22 Ex: 44162 xxx 5353259 2 xxx

10. 42 x 11. 642 x 12. 812 x

13. 1625 2 x 14. 19 2 x 15. 3649 2 x

Factor the Greatest Common Factor then factor the expression. Ex: 33292182 22 xxxx

16. 44 2 x 17. 546 2 x 18. 246 2 x

19. 322 2 x 20. 728 2 x 21. 5424 2 x

3


Factoring Trinomials To factor a quadratic expression means to write it as the product of two linear expressions.

Standard Form: cbxax 2

1. Factoring when b is positive and c is Positive

Ex: Factor 862 xx . 862 xx

)4)(2( xx Check: by multiplying.

)4)(2( xx = 8242 xxx FOIL

= 862 xx

2. Factoring when b is Negative and c is Positive

Ex: 652 xx . 652 xx

)3)(2( xx Check: )3)(2( xx = 6232 xxx

= 652 xx

3. Factoring when b is negative and c is Negative

Ex: 1032 xx . 1032 xx )5)(2( xx

4. Factoring when b is Positive and c is Negative

Ex: Factor 1872 xx . 1872 xx )9)(2( xx

Factor the trinomial.

1. 1272 xx 2. 16102 xx 3. 2452 xx

4. 202 xx 5. 982 xx 6. 21102 xx

7. 36132 xx 8. 1832 xx 9. 4062 xx

10. 3072 xx 11. 32122 xx 12. 4472 xx

13. 1452 xx 14. 1892 xx 15. 3692 xx

4


Factoring Trinomials a>1

cbxax 2 Need to find the factors of a and the factors of c.

factors of a (m and n) -first positions in each binomial(always positive)

factors of c ( p and q) -second position in each binomial

qnxpmx

Ex: 5112 2 xx factors of a 1 and 2

factors of c 1 and 5

)12)(5( xx -Arrange both sets of factors so that the middle term equals 11x.

-This process can take a couple of attempts before you find the right combination.

Check by using FOIL: 51125102)12)(5( 22 xxxxxxx

Ex: 823 2 xx factors of a 1 and 3

factors of c 1 and 8 and 2 and 4

432 xx

Check by using FOIL: 8238643432 22 xxxxxxx

Factor.

1. 5163 2 xx 2. 295 2 xx 3. 673 2 xx

4. 328 2 xx 5. 5112 2 xx 6. 743 2 xx

7. 992 2 xx 8. 372 2 xx 9. 143 2 xx

10. 8103 2 xx 11. 384 2 xx 12. 6117 2 xx

13. 968 2 xx 14. 4146 2 xx 15. 15520 2 xx

5


Solving Polynomial Equations by Factoring

Zero Product Property

Let A and B be real numbers or expressions. If 00,0 BorAthenBA .

Ex: Solve 024102 xx 064 xx Factor.

0604 xx Zero product property.

64 xx Solve for x.

Solve the equations by factoring.

1. 01072 xx 2. 01452 xx 3. 0962 xx

4. 01492 xx 5. 0442 xx 6. 062 xx

7. 40132 xx 8. 1032 xx 9. 2762 xx

10. 010173 2 xx 11. 05136 2 xx 12. 09188 2 xx

13. 372 2 xx 14. 01073 2 xx 15. 72512 2 xx

Factor out the GCF and solve.

16. 0243543 2 xx 17. 060288 2 xx 18. 02482 2 xx

6


Square Roots

Square Root of a Number If ab 2 , then b is a square root of a. bba 2

Ex: 932 therefore 39

Product Property of Radicals baab where 0a and 0b .

Ex: 52545420

Simplify the expression.

1. 16 2. 64 3. 49 4. 121

5. 12 6. 75 7. 18 8. 200

9. 32 10. 192 11. 54 12. 84

13. 72 14. 132 15. 288 16. 117

Quotient Property of Radicals b

a

b

a where 0a and 0b

Ex: 3

2

9

4

9

4

Rationalizing the Denominator No radicals are in the denominator of a fraction.

2422

Ex: Simplify2

1.

2

1

2

1

2

1 no radicals in denominator.

2

2

2

2

2

1 rationalize the denominator.

Simplify the expression.

17. 25

1 18.

64

9 19.

100

49 20.

32

18

21. 81

11 22.

4

5 23.

7

1 24.

3

2

25. 5

8 26.

14

18 27.

5

12 28.

8

27

7


Solving Quadratic Equations by finding square roots: 1. Isolate x

2 on one side of the equation.

2. Find the square root of each side.

Ex: Solve the equation. 42 x 82 x 182 2 x

42 x 82 x 92 x

2x 8ix 9x

22ix 3x

1. 12 x 2. 162 x 3. 72 x 4. 122 x

5. 812 x 6. 5005 2 x 7. 63 2 x 8. 324 2 x

9. 2203 2 x 10. 2752 2 x 11. 9307 2 x 12. 41132 2 x

Ex: Solve 36162x . 1. Isolate the squared binomial 61

2x

2. Take the square root of both sides. 612

x

3. Add/Subtract value to both sides. 61 x

61x

4. Separate into two solutions, one + and one – . 6161 xorx

13. 2522x 14. 5015

2x 15. 832

2x 16. 2824

2x

Use a calculator to solve the equation. (Round to the nearest hundredth)

17. 5734 2 x 18. 34226 2 x 19. 10142 2 x 20. 612432

x

8


Complex Number System

The Square Root of a Negative Numbers

Imaginary unit: 1i and 12 i :

If r is a positive real number, then rir .

Ex: 55 i Ex: i24 Ex: 228 i

Ex: Solve 26103 2 x . 363 2 x

122 x

12x

32ix

Simplify.

1. 64 2. 25 3. 100 4. 144

5. 27 6. 75 7. 72 8. 45

Solve.

9. 492 x 10. 813 2 x 11. 1372 x

12. 4192 2 x 13. 1442 x 14. 1332 2 x

15. 1622

x 16. 120562 x

9


The Quadratic Formula

The Quadratic Formula: The solutions of the quadratic equation 02 cbxax are:

a

acbbx

2

42

Use the quadratic formula to solve the equation. Place the equation in standard form.

1. 0342 xx 2. 012 xx 3. 01452 xx

4. 0532 2 xx 5. 0135 2 xx 6. 0386 2 xx

7. 182 xx 8. xx 619 2 9. 263 2 xx

10


Real nth Roots: Odd index : one real solution

Ex: 51253 Ex: 51253

Even index : two real solutions

Ex: 3814 Ex: solutionrealno4 81

Rational Exponents: mnmnnm aaa 1

Ex: 2739993332123

Ex: 4232323222

525152

Evaluate the expression.

1. 121 2. 3 64 3. 4 16 4. 5 243 5. 3 216

6. 4 625 7. 5 1 8. 196 9. 3 343 10. 169

11. 316 12. 23 27 13. 34 256 14. 25 1024 15. 525

Evaluate the expression.

16. 2149 17. 5132 18. 21144 19. 41256 20. 31125

21. 254 22. 348 23. 4381 24. 32512 25. 56

1

11


Rational expression: a fraction whose numerator, denominator, or both are nonzero polynomials.

Ex: 4

3

x

9

22 x

x

1

132

x

x

Simplifying Fractions: b

a

cb

ca

bc

ac

Ex:

226

6

12

6 2 x

x

xx

x

x

-Factor the numerator and denominator completely.

-cancel out like factors.

Ex: Simplify xx

xx

x

xx

32

)3(2

6

622

2

Ex: Simplify: 12

22

2

42

2

xx

xx

xx

x

Note: When you simplify rational expressions, you can only cancel out factors, not terms.

Incorrect: Ex: 66

2

2

x

x NO!!

1. x8

20 2.

4

7

63

21

x

x 3.

x

xx

36

642 3 4.

2

22

x

xx

5. xx

x

2410

1252

6.

102

252

x

x 7.

158

52

xx

x 8.

6

6112 2

x

xx

9. 152

202

2

xx

xx 10.

4

1492

23

x

xxx 11.

xxx

xx

65 23

3

12.

xxx

xxx

3662

4828423

3

13. 1

432

2

x

xx 14.

823

18112

2

xx

xx 15.

1892

151962

2

xx

xx

12

xx

xx

32

32

x

x

3

3

1

2

x

x


Answer Key

Page 1 Fractions

Odd Answers:

1. 24

19 3.

4

5 5.

15

1 7. 1 9.

15

8 11.

8

5 13.

6

5 15.

4

9 17.

3

5 19.

4

5 21. 2

23. 45

Page 2 Proportions

Odd Answers:

1. 3

4x 3.

2

7x 5.

4

3x 7.

2

11x 9. 56x 11.

3

71x 12. 3x

Page 3 Factoring Special Products

Odd Answers:

1. 875 x 3. 233 2 x 5. 326 xx 7. 144 2 xx 9. 365 2 xx 11. 88 xx

13. 4545 xx 15. 6767 xx 17. 336 xx 19. 442 xx 21. 32326 xx

Page 4 Factoring Trinomials

Odd Answers:

1. 43 xx 3. 83 xx 5. 19 xx 7. 94 xx 9. 104 xx 11. 84 xx

13. 27 xx 15. 123 xx

Page 5 Factoring Trinomials a > 1

Odd Answers:

1. 513 xx 3. 323 xx 5. 512 xx 7. 332 xx 9. 113 xx

11. 3212 xx 13. 3234 xx 15. 1345 xx

Page 6 Solving Polynomial Equations by Factoring

1. 5,2x 2. 7,2 x 3. rootdoublex 3 4. 7,2x 5. ..2 rdx 6. 2,3x

7. 5,8 x 8. 5,2x 9. 3,9x 10. 5,3

2x 11.

3

5,

2

1x 12.

4

3,

2

3x

13. 3,2

1x 14. 1,

3

10x 15.

4

1,

3

7x 16. ..9 rdx 17. 5,

2

3x 18. 6,2 x

Page 7 Square Roots

Odd Answers:

1. 4 3. 7 5. 32 7. 23 9. 24 11. 63 13. 26 15. 212

17. 5

1 18.

8

3 19.

10

7 20.

4

3 21.

9

11 22.

2

5 23.

7

7 24.

3

6 25.

5

102 26.

7

73

27. 5

152 28.

4

63


Page 8 Solving Quadratic Equations by Finding Square Roots

1. 1x 2. 4x 3. 7x 4. 32x 5. 9x 6. 10x 7. 2x

8. 22x 9. 6x 10. 4x 11. 3x 12. 14x 13. 7,3x

14. 101x 15. 5,1x 16. 72x 17. 87.3x 18. 41.1x 19. 46.3x

20. 55.1,45.6 x

Page 9 Complex Number System

Odd Answers:

1. i8 3. i10 5. 33i 7. 26i

9. ix 7 10. 33ix 11. 52ix 12. ix 5 13. 23ix 14. 22ix

15. ix 42 16. 525 ix

Page 10 The Quadratic Formula

1. 3,1x 2. 2

51x 3. 7,2x 4.

4

313 ix

5.

10

293x 6.

6

2

3

2 ix

7. 154x 8. 3

1x 9.

3

31x

Page 11 Rational Exponents

Odd Answers:

1. 11 3. 2 5. –6 7. –1 9. 7 11. 64 13. 64 15. 3,125 17. 2 19. 4 21. 32

23. 27 25. 1

Page 12 Rational Expressions

1. x2

5 2.

3

3x 3.

6

7 2x 4. x 5.

x2

1 6.

2

5x 7.

3

1

x 8. 12 x 9.

3

4

x

x

10.

2

7

x

xx 11.

6

1

x

xx 12.

6

42

x

x 13.

1

43

x

x 14.

43

9

x

x 15.

6

53

x

x

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Trigonometry Algebra 2 Review - HASD