Unit P: PrerequisiteUnit P: Prerequisite

P.1 Real NumbersP.1 Real NumbersP.2 Exponents and RadicalsP.2 Exponents and RadicalsP.3 Polynomials and FactoringP.3 Polynomials and FactoringP.4 Rational ExpressionsP.4 Rational ExpressionsP.5 Solving EquationsP.5 Solving EquationsP.6 Solving InequalitiesP.6 Solving InequalitiesP.7 Errors and the Algebra of CalculusP.7 Errors and the Algebra of CalculusP.8 Graphical Representation of DataP.8 Graphical Representation of Data

Warm-UpWarm-Up

1.1. What is an integer?What is an integer?

2.2. Evaluate |-3 - 4|Evaluate |-3 - 4|

3.3. What is the identity element for addition?What is the identity element for addition?

4.4. ⅜ ⅜ ÷ ⅔ = ____÷ ⅔ = ____

P.1 Real NumbersP.1 Real Numbers

ObjectiveObjective – To order real numbers, use – To order real numbers, use inequalities, and evaluate algebraic inequalities, and evaluate algebraic expressions.expressions.

Real NumbersReal Numbers Ordering Real NumbersOrdering Real Numbers Absolute Value and DistanceAbsolute Value and Distance Algebraic ExpressionsAlgebraic Expressions Basic Rules of AlgebraBasic Rules of Algebra

Real NumbersReal Numbers

What are the Real Numbers?What are the Real Numbers?

How are real numbers represented How are real numbers represented graphically?graphically?

Ordering Real NumbersOrdering Real Numbers

Ordering Real NumbersOrdering Real Numbers

Ordering Real NumbersOrdering Real Numbers

Absolute Value and DistanceAbsolute Value and Distance

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

Absolute Value and DistanceAbsolute Value and Distance

Algebraic ExpressionsAlgebraic Expressions

What is an Algebraic Expression?What is an Algebraic Expression?

An algebraic expression is a collection of letters An algebraic expression is a collection of letters (variables) and real numbers (constants) combined using (variables) and real numbers (constants) combined using the operations of addition, subtraction, multiplication, the operations of addition, subtraction, multiplication, division, and exponentiation.division, and exponentiation.

22x - x - 33Variable term

Constant TermCoefficient

Algebraic ExpressionsAlgebraic Expressions

To evaluate an algebraic expression, To evaluate an algebraic expression, substitute numerical values for each of the substitute numerical values for each of the variables in the expression.variables in the expression.

Value ofValue of Value ofValue of

ExpressionExpression VariableVariable SubstituteSubstituteExpressionExpression

-3x+5 -3x+5 for x=for x=33 -3(-3(33)+5)+5 -9+5=-4-9+5=-4

3x3x²+2x-1²+2x-1 for x=for x=-1-1 3(3(-1-1)²+2()²+2(-1-1)-1)-1 3-2-1=03-2-1=0

Basic Rules of AlgebraBasic Rules of Algebra

Basic Rules of AlgebraBasic Rules of Algebra

Basic Rules of AlgebraBasic Rules of Algebra

Basic Rules of AlgebraBasic Rules of Algebra

Basic Rules of AlgebraBasic Rules of Algebra

Finally…those effecting FACTORINGFinally…those effecting FACTORING

If If aa, , bb, and , and cc are integers such that are integers such that ab=cab=c, then , then aa and and bb are are factorsfactors of of cc..

A A Prime numberPrime number is an integer that has exactly two positive is an integer that has exactly two positive factors, itself and 1. Examples are 2, 3, 5, 7 and 11.factors, itself and 1. Examples are 2, 3, 5, 7 and 11.

Composite numbersComposite numbers are those numbers that can be are those numbers that can be written as the product of two or more prime numbers.written as the product of two or more prime numbers.

The number The number 11 is neither prime nor composite. is neither prime nor composite.

Basic Rules of AlgebraBasic Rules of Algebra

This brings us to…..This brings us to…..

The Fundamental Theorem of ArithmeticThe Fundamental Theorem of Arithmetic

Every positive integer greater than 1 can Every positive integer greater than 1 can be written as the product of prime be written as the product of prime numbers in precisely one way numbers in precisely one way (disregarding order). This is called (disregarding order). This is called prime prime factorization.factorization.

SynthesisSynthesis

1.1. Consider |Consider |u+vu+v| and || and |uu| + || + |vv|.|.a)a) Are the values of the expressions always equal? Are the values of the expressions always equal?

If not, under what conditions are they unequal? If not, under what conditions are they unequal?b)b) If the two expressions are not equal for certain If the two expressions are not equal for certain

values of values of uu and and vv, is one of the expressions , is one of the expressions always greater than the other? Explain.always greater than the other? Explain.

2.2. Is there a difference between saying that a Is there a difference between saying that a real number is positive and saying that a real number is positive and saying that a real number is nonnegative? Explain.real number is nonnegative? Explain.

3.3. Because every even number is divisible by Because every even number is divisible by 2, is it possible that there exist any even 2, is it possible that there exist any even prime numbers? Explain.prime numbers? Explain.

Warm-UpWarm-Up

1.1. Simplify Simplify

2.2. Write in Scientific Notation 39,000,000Write in Scientific Notation 39,000,000

3.3. Simplify Simplify

4.4. Simplify Simplify

2503

6 1

3

2x y

y

FHG

IKJ

9

9

2

3

1

6

P.2 Exponents and RadicalsP.2 Exponents and Radicals

Objective – To use properties of exponents, Objective – To use properties of exponents, radicals and rational exponents in order to radicals and rational exponents in order to simplify algebraic expressions. To use scientific simplify algebraic expressions. To use scientific notation.notation.

ExponentsExponents Scientific NotationScientific Notation Radicals and Their PropertiesRadicals and Their Properties Simplifying RadicalsSimplifying Radicals Rationalizing Denominators and NumeratorsRationalizing Denominators and Numerators Rational ExponentsRational Exponents

ExponentsExponents

Repeated multiplication can be written in Repeated multiplication can be written in exponential form.exponential form.

(2x)(2x)(2x) = (2x)(2x)(2x)(2x) = (2x)³³

Where 2x is the base and 3 is the exponent.Where 2x is the base and 3 is the exponent.

ExponentsExponents

ExponentsExponents

Additional Examples Additional Examples

25

1FHGIKJ

4 1x

3

2 2x

( ) 3 5

35

RadicalsRadicals

Exponential FormExponential Form

Radical FormRadical Form

a bn

ab if n is even

b if n is oddn RST| |

SimplifySimplify

16

6

2 3 4

5 2

3x w z

x wz

FHG

IKJ

r s 33 63