Algebra Review

Warm-up (3 m)• Multiply:1. 4x2(7x3 - 6x2 + 12x - 10) 2. (3x2 - 5)(x +

4)

• Factor:3. x3 – 64x 4. 9x2 – 9x – 4

We’re going to review the following skills for the next unit:

• Multiplying Polynomials• Factoring Polynomials• Simplifying Rational Expressions• Multiplying Rational Expressions• Dividing Rational Expressions• Adding and Subtracting Rational Expressions

Multiplying PolynomialsDistribute

• Multiply each term inside the parentheses by the term outside the parentheses.

3x5(x7 – 2x4 + 11x)

FOIL

• First – Outer – Inner – Last

(3x + 5)(x – 7)

Multiplying Polynomials, cont.

• Works well when you multiply anything larger than a binomial and a binomial.– Example: (3x2 – x + 1)(x2 + 2x – 3)

• Very similar to long multiplication by hand.

Vertical Multiplication

Example• (7x2 – 5x +6)(2x – 1)

Example• (3x2 – x + 1)(x2 + 2x – 3)

Multiplying with Trigonometric Functions

• Exactly the same as multiplying without trigonometric functions.

)3xtan2x(tanxtan 2

Your Turn:

• Multiply problems 1 – 10 in the Algebra Review packet

1. 2.

3. 4.

5.

6. 7.

8. 9.

10.

Factoring• Remember, there are four types of factoring

that we reviewed at the beginning of the semester:– Leading Coefficient = 1 (“Regular” Factoring)– Leading Coefficient ≠ 1 (Box Method or Welsh

Method)– Greatest Common Factor (GCF)– Difference of Squares

Leading Coefficient = 1

x2 – 7x + 10

Leading Coefficient ≠ 1

3x2 – 11x – 20

Greatest Common Factor

4x4 – 40x3 + 8x2

Difference of Squares

81x4 – 100

Factoring with Trigonometric Functions

• Exactly the same as factoring without trigonometric functions.

xcos2xcosxsin 2 4xsin2

Your Turn:

• Factor problems 11 – 24 in the Algebra Review packet.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

Warm-up (3 m)1. Multiply: 2. Factor:

• Find the reciprocals of the numbers below:

3. 4. 7

)5xtan2)(2xtan4x(tan2 xsecxcot 44

y11x6

Seek and Solve!!!

Show all your work on a piece of paper. I’m collecting it for a

classwork grade.

Simplifying Rational Expressions

• You can only cancel factors that are separated by multiplication!!!

75

7x5x

7x2x

)9x)(7x()9x)(2x(

Wrong!!! Right!!!

Simplifying Rationals, cont.• You can also reduce factors – as long as

they’re separated by multiplication.

2x10x5

27

42

mg4mg6

Simplifying Rationals, cont.

1. Factor the numerator and the denominator2. Optional – Identify the factors in the

numerator and the denominator. 3. Cancel common factors in the numerator and

the denominator.

Example

Factors in Numerator

Factors in Denominator

12x22x4x48x4x2

2

23

Example

Factors in Numerator

Factors in Denominator

xsinxcotxsin1xcot2

Your Turn:

• Simplify problems 25 – 32 in the Algebra Review packet. Remember to factor the numerator and the denominator first, AND you can only cancel factors separated by multiplication.

25. 26.

27. 28.

29. 30.

31. 32.

Warm-up (3 m)1. Simplify:

xsinx2xcosx2xsin6xcos622

44

xsinx2xcosx2xsin6xcos622

44

Homework Review

Multiplying Rational Expressions

1. Factor the numerator and the denominator.2. Cancel and/or cross cancel any common

factors that are separated by multiplication.3. Optional – Rewrite the simplified fractions.4. Multiply across. (Multiply the numerators

together and the denominators together.)

Example

10x5x

x516x8 3

Example

5xcosxcos3xcos

9xcos25cos10xcos 2

2

2

Your Turn

• Multiply problems 33 – 38 in the Algebra Review packet. Simplify your answers.

33. 34.

35. 36.

37. 38.

What About…?

3

4

4

3

xx4xx

xxx4x

Dividing Rational Expressions

• Division is the same thing as multiplication by the reciprocal!

510

31

21

Dividing Rationals, cont.

1. Rewrite the division as multiplication by the reciprocal.

2. Factor the numerator and the denominator.3. Cancel and/or cross cancel any common

factors separated by multiplication.4. Multiply across.

Example

9

2

5

y16x3y8x9

Example

3

8

x2x

x10x5

Your Turn:

• Divide problems 39 – 48 in the Algebra Review packet.

39. 40.

41. 42.

43. 44.

45. 46.

Adding and Subtracting Rational Expressions

• If the fractions have the same denominator, add or subtract the numerators. (Make sure to distribute the subtraction sign!!!)

• Simplify the fraction is possible.

Examples

7x21x6x

7x5x 2

2xsin5xsin2

2xsin4xsin

Adding and Subtracting Rational Expressions, cont.

• If the fractions have the different denominators,1. Factor the numerator and the denominator.2. Simplify each fraction individually if possible.3. Compare the denominators of each fraction. Identify

the “missing” factors from each fraction. (Finding the Least Common Denominator)

4. Multiply each fraction by 1. Rewrite 1 as factors"gsinmis"factors"gsinmis"

Adding and Subtracting Rational Expressions, cont.

5. Multiply across.6. Combine all fractions into one fraction.7. Simplify the numerator.8. Factor the numerator if possible.9. Simplify/reduce the fraction if possible.

Examples in Smart Board File

Example

4x7x

2x5x

Example

15xtan1xtan

10xtan3xtan

Example

4xsin2xsin212xsin4

8xsin2xsin12xsinxsin

22

2

Your Turn:

• Add or subtract problems 49 – 62 in the Algebra Review packet.

49. 50.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

61. 62.

Activity!!!