Date: Section P.2: Exponents and Radicals. Properties of Exponents: Example #1: Simplify. a.) 3 4. b.) 2. c.) 3 4. d.) Example #2: Simplify. b.) a.

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1 Properties of Exponents: Section P.2: Exponents and Radicals Date: Example #1: Simplify. a.) 3 4 b.) 2 c.) 34 d.) Example #2: Simplify. a.) b.) c.) d.) 1

2 Square Root: Principal n th Root: Example #3: Simplify. a.) 49 = b.) 49 c.) = d.) 32 = e.) 8 2

3 Properties of Radicals: Example #4: Simplify. a.) 8 2 = b.) 5 c.) 3

4 Simplifying Radicals: An expression involving radicals is in simplest form when the following conditions are satisfied: 1.) All possible factors have been removed from the radical. 2.) All fractions have radical-free denominators (rationalizing the denominator not in this class!) 3.) The index of the radical is reduced. Example #5: Simplifying Even Roots 4 a.) 48 b.) 75x 3 4 c.) (5x) 4 4

5 Example #6: Simplifying Odd Roots 3 a.) 24 b.) 3 24a 4 c.) 3 40x 6 Rational Exponents: Definition of Rational Exponents: If a is a real number and n is a positive integer such that the principal nth root of a exists, we define 1 to be a n a 1 n = 5

6 If m is a positive integer that has no common factor with n, then a m n ( ) and a Example #7: Changing from Radical to Exponential Form a.) 3 = a 1 n m = n a m m n = ( a m ) 1 n = n a m b.) ( 3xy) 5 c.) 2x 4 x 3 Example #8: Changing from Exponential to Radical Form 3 ( a.) x 2 + y 2 ) 2 6

7 b.) 2y z 3 2 c.) a d.) x 0.2 Example #9: Simplifying with Rational Exponents ( a.) 27 2 ) 6 b.) ( 32) 4 5 7

8 c.) 5x 5 3 3x d.) a 3 3 e.) 125 f.) ( 2x 1) ( 2x 1) 3 8

9 Example #10: Combining Radicals a.) b.) 3 16x 3 54x 4 9

10 Section P.3: Polynomials and Factoring Date: Polynomials: In standard form, a polynomial is written with descending powers of x. The highest exponent in the polynomial is the degree, and the number in front of that term is the leading coefficient. The number in the polynomial without a variable is called the constant term. Example #1: Writing Polynomials in Standard form. 4x 2 5x x Example #2: Sums and Differences of Polynomials (7x 4 x 2 4x + 2) (3x 4 4x 2 + 3x) 10

11 Example #3: Multiplying Polynomials The FOIL Method (3x 2)(5x + 7) Example #4: The Product of Two Trinomials (x + y 2)(x + y + 2) Example #5: Removing Common Factors a.) 3x 3 + 9x 2 b.) (x 2)(2x) + (x 2)(3) 11

12 Example #6: Removing a Common Factor First 3 12x 2 Example #7: Factoring the Difference of Two Squares a.) (x + 2)2 y 2 b.) 16x 4 81 Example #8: Factoring Perfect Square Trinomials a.) 16x 2 + 8x +1 b.) x 2 10x

13 Example #11: Factoring a Trinomial: Leading Coefficient Is 1 x 2 7x +12 Example #12: Factoring a Trinomial: Leading Coefficient Is Not 1 2x 2 + x 15 Example #13: Factoring by Grouping x 3 2x 2 3x

14 Domain of an Algebraic Expression: Section P.4: Fractional Expressions Date: The set of real numbers for which an algebraic expression is defined is the domain. Example #1: Finding the Domain of an Algebraic Expression a.) The domain of the polynomial: 2x 3 + 3x + 4 is b.) The domain of the radical expression x 2 is c.) The domain of the expression x + 2 x 3 is 14

15 Example #2: Reducing a Rational Expression Write x 2 + 4x 12 3x 6 in reduced form. Simplifying Rational Expressions: Example #3: Reducing Rational Expressions a.) x 3 4 x x 2 + x 2 b.) 12 + x x 2 2x 2 9x

16 Operations with Rational Expressions: Example #4: Multiplying Rational Expressions 2x 2 + x 6 x x 5 x 3 3x 2 + 2x 4 x 2 6x Example #5: Dividing Rational Expressions x 3 8 x 2 4 x 2 + 2x + 4 x

17 Example #6: Subtracting Rational Expressions x x 3 2 3x + 4 Example #7: Combining Rational Expressions: The LCD Method 3 x 1 2 x + x + 3 x

18 Compound Fractions: Example #8: Simplifying a Compound Fraction 2 x x 1 Example #9: Simplifying an Expression with Negative Exponents 3 1 x(1 2x) (1 2x) 18

19 Example #10: Simplifying a Compound Fraction (4 x 2 ) x 2 (4 x 2 ) x 2 19

20 Date: Section P.5: Solving Equations Linear Equations: Example #1: Solving a Linear Equation Solve 3x 6 = 0 Example #2: An Equation Involving Fractional Expressions Solve x 3 + 3x 4 = 2 Example #3: An Equation with an Extraneous Solution Solve 1 x 2 = 3 x + 2 6x x

21 Example #4: Solving Quadratic Equations by Factoring a.) 2x 2 + 9x + 7 = 3 b.) 6x 2 3x = 0 Example #5: Extracting Square Roots a.) 4 x 2 = 12 b.) ( x 3) 2 = 7 21

22 Example #6: The Quadratic Formula: Two Distinct Solutions Use the Quadratic Formula to solve: x 2 + 3x = 9 Example #7: The Quadratic Formula: One Repeated Solution Use the Quadratic Formula to solve: 8x 2 24 x +18 = 0 22

23 Polynomial Equations of Higher Degree: Example #8: Solving a Polynomial Equation by Factoring Solve 3x 4 = 48 x 2 Example #9: Solving a Polynomial Equation by Factoring Solve x 3 3x 2 3x + 9 = 0 Radical Equations: Example #10: Solving an Equation Involving a Rational Exponent Solve 4 x = 0 23

24 Example #11: Solving an Equation Involving a Radical Solve 2x + 7 x = 2 Absolute Value Equations: Example #12: Solving an Equation Involving Absolute Value Solve x 2 3x = 4 x