Algebra Review. Warm-up (3 m) Multiply: 1. 4x 2 (7x 3 - 6x 2 + 12x - 10) 2. (3x 2 - 5)(x + 4)...
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Transcript of Algebra Review. Warm-up (3 m) Multiply: 1. 4x 2 (7x 3 - 6x 2 + 12x - 10) 2. (3x 2 - 5)(x + 4)...
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
Multiplying with Trigonometric Functions
• Exactly the same as multiplying without trigonometric functions.
)3xtan2x(tanxtan 2
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
Factoring with Trigonometric Functions
• Exactly the same as factoring without trigonometric functions.
xcos2xcosxsin 2 4xsin2
Warm-up (3 m)1. Multiply: 2. Factor:
• Find the reciprocals of the numbers below:
3. 4. 7
)5xtan2)(2xtan4x(tan2 xsecxcot 44
y11x6
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.
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.
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.)
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.
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.
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.