warm up multiply: 1. 4x 2 (7x 3 - 6x 2 + 12x - 10) 2. (3x 2 - 5)(x + 4) factor: 3. x 3 – 64x4. 9x...
DESCRIPTION
Multiplying Polynomials Distribute Multiply each term inside the parentheses by the term outside the parentheses. 3x 5 (x 7 – 2x x) FOIL First – Outer – Inner – Last (3x + 5)(x – 7) EQ: How do I perform algebraic operations involving trigonometric expressions?TRANSCRIPT
Warm Up• Multiply:1. 4x2(7x3 - 6x2 + 12x - 10) 2. (3x2 - 5)(x +
4)
• Factor:3. x3 – 64x 4. 9x2 – 9x – 4
EQ: How do I perform algebraic operations involving trigonometric expressions?
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
EQ: How do I perform algebraic operations involving trigonometric 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)
EQ: How do I perform algebraic operations involving trigonometric expressions?
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
EQ: How do I perform algebraic operations involving trigonometric expressions?
Example• (7x2 – 5x +6)(2x – 1)
EQ: How do I perform algebraic operations involving trigonometric expressions?
Example• (3x2 – x + 1)(x2 + 2x – 3)
EQ: How do I perform algebraic operations involving trigonometric expressions?
Multiplying with Trigonometric Functions
• Exactly the same as multiplying without trigonometric functions.
)1x)(sin1x(sin )3xtan2x(tanxtan 2
EQ: How do I perform algebraic operations involving trigonometric expressions?
Your Turn:
• Multiply problems 1 – 10 in the Algebra Review packet
EQ: How do I perform algebraic operations involving trigonometric expressions?
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
EQ: How do I perform algebraic operations involving trigonometric expressions?
Leading Coefficient = 1
x2 – 7x + 10
EQ: How do I perform algebraic operations involving trigonometric expressions?
Leading Coefficient ≠ 1
3x2 – 11x – 20
EQ: How do I perform algebraic operations involving trigonometric expressions?
Greatest Common Factor
4x4 – 40x3 + 8x2
EQ: How do I perform algebraic operations involving trigonometric expressions?
Difference of Squares
81x4 – 100
EQ: How do I perform algebraic operations involving trigonometric expressions?
Factoring with Trigonometric Functions
• Exactly the same as factoring without trigonometric functions.
xcos2xcosxsin 2 4xsin2
EQ: How do I perform algebraic operations involving trigonometric expressions?
Your Turn:
• Factor problems 11 – 24 in the Algebra Review packet.
EQ: How do I perform algebraic operations involving trigonometric expressions?
1. Multiply: 2. Factor:
• Find the reciprocals of the numbers below:
3. 4. 7
)5xtan2)(2xtan4x(tan2 xsecxcot 44
y11x6
EQ: How do I perform algebraic operations involving trigonometric expressions?
Simplifying Rational Expressions
• You can only cancel factors that are separated by multiplication!!!
75
7x5x
7x2x
)9x)(7x()9x)(2x(
Wrong!!! Right!!!
EQ: How do I perform algebraic operations involving trigonometric expressions?
Simplifying Rationals• You can also reduce factors – as long as
they’re separated by multiplication.
2x10x5
27
42
mg4mg6
EQ: How do I perform algebraic operations involving trigonometric expressions?
Simplifying Rationals
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.
EQ: How do I perform algebraic operations involving trigonometric expressions?
Example
Factors in Numerator
Factors in Denominator
12x22x4x48x4x2
2
23
EQ: How do I perform algebraic operations involving trigonometric expressions?
ExampleFactors in Numerator
Factors in Denominator
xsinxcotxsin1xcot2
EQ: How do I perform algebraic operations involving trigonometric expressions?
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.
EQ: How do I perform algebraic operations involving trigonometric expressions?
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.)
EQ: How do I perform algebraic operations involving trigonometric expressions?
Example
10x5x
x516x8 3
EQ: How do I perform algebraic operations involving trigonometric expressions?
Example
5xcosxcos3xcos
9xcos25cos10xcos 2
2
2
EQ: How do I perform algebraic operations involving trigonometric expressions?
Your Turn
• Multiply problems 33 – 38 in the Algebra Review packet. Simplify your answers.
EQ: How do I perform algebraic operations involving trigonometric expressions?
Warm Up1. Simplify:
2. Multiply:25xsin24xsin4
12xsin8xsin10xsin2
22
xsinx2xcosx2xsin6xcos622
44
EQ: How do I perform algebraic operations involving trigonometric expressions?
xsinx2xcosx2xsin6xcos622
44
EQ: How do I perform algebraic operations involving trigonometric expressions?
25xsin24xsin4
12xsin8xsin10xsin2
22
EQ: How do I perform algebraic operations involving trigonometric expressions?
Dividing Rational Expressions
• Division is the same thing as multiplication by the reciprocal!
510
31
21
EQ: How do I perform algebraic operations involving trigonometric expressions?
Dividing Rationals
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.
EQ: How do I perform algebraic operations involving trigonometric expressions?
Example
9
2
5
y16x3y8x9
EQ: How do I perform algebraic operations involving trigonometric expressions?
Example
3
8
x2x
x10x5
EQ: How do I perform algebraic operations involving trigonometric expressions?
Your Turn:
• Divide problems 39 – 48
EQ: How do I perform algebraic operations involving trigonometric expressions?
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.
EQ: How do I perform algebraic operations involving trigonometric expressions?
Examples
7x21x6x
7x5x 2
2xsin5xsin2
2xsin4xsin
EQ: How do I perform algebraic operations involving trigonometric expressions?
Adding and Subtracting Rational Expressions
• 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"
EQ: How do I perform algebraic operations involving trigonometric expressions?
Adding and Subtracting Rational Expressions
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.
EQ: How do I perform algebraic operations involving trigonometric expressions?
Example
4x7x
2x5x
EQ: How do I perform algebraic operations involving trigonometric expressions?
Example
15xtan1xtan
10xtan3xtan
EQ: How do I perform algebraic operations involving trigonometric expressions?
Example
4xsin2xsin212xsin4
8xsin2xsin12xsinxsin
22
2
EQ: How do I perform algebraic operations involving trigonometric expressions?
Your Turn:
• Add or subtract problems 49 – 62 in the Algebra Review packet.
EQ: How do I perform algebraic operations involving trigonometric expressions?