warm up multiply: 1. 4x 2 (7x 3 - 6x 2 + 12x - 10) 2. (3x 2 - 5)(x + 4) factor: 3. x 3 – 64x4. 9x...

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?