algebra review
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Algebra Review. Warm-up (3 m). Multiply: 1. 4x 2 (7x 3 - 6x 2 + 12x - 10) 2. (3x 2 - 5)(x + 4) Factor: 3. x 3 – 64x4. 9x 2 – 9x – 4. We’re going to review the following skills for the next unit:. Multiplying Polynomials Factoring Polynomials - PowerPoint PPT PresentationTRANSCRIPT
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.
Warm-up (4 m)1. Multiply:
25xsin24xsin4
12xsin8xsin10xsin2
22
25xsin24xsin4
12xsin8xsin10xsin2
22
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!!!