1. Unit P: Prerequisite P.1 Real Numbers P.2 Exponents and Radicals P.3 Polynomials and Factoring P.4 Rational Expressions P.5 Solving Equations P.6 Solving Inequalities P.7 Errors and the Algebra of Calculus P.8 Graphical Representation of Data

2. Warm-Up 1. What is an integer? 2. Evaluate |-3 - 4| 3. What is the identity element for addition? 4. ⅜ ÷ ⅔ = ____

3. P.1 Real Numbers Objective – To order real numbers, use inequalities, and evaluate algebraic expressions.  Real Numbers  Ordering Real Numbers  Absolute Value and Distance  Algebraic Expressions  Basic Rules of Algebra

4. Real Numbers Real Numbers What are the Real Numbers? How are real numbers represented graphically?

5. Ordering Real Numbers

9. Absolute Value and Distance The Absolute Value of a real number is its magnitude, or the distance between the origin and the point representing the real number on the real number line.

10. Absolute Value and Distance

11. Algebraic Expressions What is an Algebraic Expression? An algebraic expression is a collection of letters (variables) and real numbers (constants) combined using the operations of addition, subtraction, multiplication, division, and exponentiation. An algebraic expression is a collection of letters (variables) and real numbers (constants) combined using the operations of addition, subtraction, multiplication, division, and exponentiation. 2x - 3 Variable term Constant Term Coefficient

12. Algebraic Expressions To evaluate an algebraic expression, substitute numerical values for each of the variables in the expression. Value ofValue of Value ofValue of ExpressionVariableSubstituteExpression -3x+5 for x=3-3(3)+5-9+5=-4 3x²+2x-1for x=-13(-1)²+2(-1)-13-2-1=0

13. Basic Rules of Algebra

18. Finally…those effecting FACTORING If a, b, and c are integers such that ab=c, then a and b are factors of c. A Prime number is an integer that has exactly two positive factors, itself and 1. Examples are 2, 3, 5, 7 and 11. Composite numbers are those numbers that can be written as the product of two or more prime numbers. The number 1 is neither prime nor composite.

19. Basic Rules of Algebra This brings us to….. The Fundamental Theorem of Arithmetic Every positive integer greater than 1 can be written as the product of prime numbers in precisely one way (disregarding order). This is called prime factorization.

20. Synthesis 1. Consider |u+v| and |u| + |v|. a) Are the values of the expressions always equal? If not, under what conditions are they unequal? b) If the two expressions are not equal for certain values of u and v, is one of the expressions always greater than the other? Explain. 2. Is there a difference between saying that a real number is positive and saying that a real number is nonnegative? Explain. 3. Because every even number is divisible by 2, is it possible that there exist any even prime numbers? Explain.

21. Warm-Up 1. Simplify 2. Write in Scientific Notation 39,000,000 3. Simplify 4. Simplify

22. P.2 Exponents and Radicals Objective – To use properties of exponents, radicals and rational exponents in order to simplify algebraic expressions. To use scientific notation.  Exponents  Scientific Notation  Radicals and Their Properties  Simplifying Radicals  Rationalizing Denominators and Numerators  Rational Exponents

23. Exponents Repeated multiplication can be written in exponential form. (2x)(2x)(2x) = (2x)³ Where 2x is the base and 3 is the exponent.

24. Exponents

25. Exponents Additional Examples

26. Radicals Exponential Form Radical Form

31. Simplify