COLLEGE ALGEBRA Introduction P1 The Real Number System P2 Integer and Rational Number Exponents P3 Polynomials.

COLLEGE ALGEBRAIntroduction

P1 The Real Number System

P2 Integer and Rational Number Exponents

P3 Polynomials

Introduction• Welcome!• Addendum• Quarter Project• Wikispaces website:

• http://msbmoorheadmath.wikispaces.com/

• Questions?!

P1 The Real Number System• Bonus opportunity for the beginning of P1 on the

Wikispaces site!

• Number system• Prime/Composite Numbers• Absolute Value• Exponential Notation• Order of Operations

P1 - Evaluate• To evaluate an expression, replace the variables by their

given values and then use the Order of Operations.• when x = 2 and y = -3

P1 - EvaluateYou try!

Evaluate when x = 3, y = -2 and z = -4

P1 – Properties of AdditionClosure a + b is a unique number

Commutative a + b = b + a

Associative (a + b) + c = a + (b + c)

Identity a + 0 = 0 + a = a

Inverse a + (-a) = (-a) + a = 0

P1 – Properties of MultiplicationClosure ab is a unique number

Commutative ab = ba

Associative (ab)c = a(bc)

Identity a·1 = 1·a = a

Inverse

P1 – Property IdentificationWhich property do each of the following use?

P1 – Property IdentificationWhich property do each of the following use?

P1 – Property IdentificationWhich property do each of the following use?

P1 – Property IdentificationWe use properties to simplify:

First we will use the Commutative Property

Then we will use the Associative Property

P1 – Property IdentificationWe use properties to simplify:

We will use the Distributive Property

P1 – Property IdentificationUse properties to simplify:

P1 – Property IdentificationUse properties to simplify:

P1 – Properties of Equality

Reflexive a = a

Symmetric If a = b, then b = a

Transitive If a = b and b = c, then a =c

Substitutional If a = b, then a may be replaced by b in any expression that involves a.

P1 – Properties of EqualityIdentify which property of equality each equation has:

P1 – Properties of EqualityIdentify which property of equality each equation has:

P1

Time for a break!

P2 – Integer ExponentsRemember….

. Multiplied n times.

So, ,

Be careful….

P2 – Integer ExponentsIf b ≠ 0 and n is a natural number, then

and

Examples:

P2 – Integer ExponentsExamples:

You try:

P2 – Integer ExponentsYou try:

P2 – Properties of Exponents

Product

Quotient where b ≠ 0

Power

where b ≠ 0

P2 – Properties of ExponentsSimplify:

Simplify:

P2 – Properties of ExponentsSimplify:

Simplify:

P2 – Properties of ExponentsSimplify:

P2 – Properties of ExponentsYou Try:

P2 – Properties of ExponentsYou Try:

P2 – Scientific NotationA number written in Scientific Notation has the form:

Where n is an integer and

• For numbers greater than 10 move the decimal to the right of the first digit, n will be the number of places the decimal place was moved

7, 430, 000

P2 – Scientific NotationFor numbers less than 10 move the decimal to the right of the first non-zero digit, n will be negative, and its absolute value will equal the number of places the decimal place was moved

0.00000078

P2 – Scientific Notation

P2 – Scientific NotationDivide:

P2 – Rational Exponents and RadicalsIf n is an even positive integer and b ≥ 0, then is the nonnegative real number such that

If n is an odd positive integer, then is the real number such that

because

P2 – Rational Exponents and RadicalsExamples:

because

because

because

because

P2 – Rational Exponents and RadicalsExamples:

However…

( is not a real number because

If n is an even positive integer and b < 0, then is a complex number….we will get to that later…

P2 – Rational Exponents and RadicalsFor all positive integers m and n such that m/n is in simplest form, and fro all real numbers b for which is a real number.

Example:

P2 – Rational Exponents and Radicals

Example:

P2 – Rational Exponents and RadicalsSimplify:

P2 – Rational Exponents and RadicalsYou Try:

P2 –RadicalsRadicals are expressed by , are also used to denote roots. The number b is the radicand and the positive integer n is the index of the radical.

If n is a positive integer and b is a real number such that is a real number, then

If the index equals 2, then the radical also known as the principle square root of b.

P2 –RadicalsFor all positive integers n, all integers m and all real numbers b such that is a real number,

This helps us switch between exponential form and radical expressions

P2 –RadicalsWe can evaluate…

Try on our calculator!

http://web2.0calc.com/

P2 –RadicalsIf n is an even natural number and b is a real number, then

If n is an odd natural number and b is a real number, then

P2 –Radical PropertiesIf n and m are natural numbers and a and b are positive real numbers, then…

Product

Quotient

Index

P2 –Radical PropertiesIf n and m are natural numbers and a and b are positive real numbers, then…

Product

Quotient

Index

P2 –RadicalsHow do we know if our expression is in simplest form?

1. The radicand contains only powers less than the index.

2. The index of the radical is as small as possible.

3. The denominator has been rationalized. Such that no radicals occur in the denominator.

4. No fractions occur under the radical sign.

P2 –RadicalsSimplify:

P2 –RadicalsSimplify:

P2 –RadicalsLike radicals have the same radicand and the same index…

P2 –RadicalsSimplify:

P2 –RadicalsMultiply:

P2 –RadicalsYou Try:

P2 –RadicalsTo Rationalize the Denominator of a fraction means to write the fraction in an equivalent form that does not involve any radicals in the denominator. To do this we multiply the numerator and denominator of the radical expression by an expression that will cause the radicand in the denominator to be a perfect root of the index…

Let’s take a look…

P2 –RadicalsExample:

P2 –RadicalsYou Try:

where y > 0

P2 –RadicalsExample:

P2 –RadicalsYou Try:

, where x > 0

P2 - Radicals• LET’S TAKE A BREAK!

P3 - PolynomialsA monomial is a constant, a variable, or the product of a constant and one or more variables with the variables having nonnegative integer exponents….

Coefficient is the number located directly in front of a variable.

The degree of a monomial is the sum of the exponents of the variables.

-8 7y

z

P3 - PolynomialsA polynomial is the sum of a finite number of monomials. Each monomial is called a term of the polynomial. The degree of a polynomial is the greatest of the degrees of the terms.

A binomial is a polynomial with two terms.

A trinomial is a polynomial with three terms.

**We always write our polynomials in descending order according to the largest exponent…

P3 - PolynomialsExample: Add

P3 - PolynomialsExample: Multiply

P3 - PolynomialsExample: Multiply using FOIL – For BINOMIALS ONLY

P3 - PolynomialsSpecial Forms –

These are for your reference, you do not have to use the special form rules, you can simply multiply manually.

P3 - PolynomialsExample:

P3 - PolynomialsExample: Evaluate the polynomial

for x = -4

P3 - PolynomialsExample: The number of singles tennis matches that can be played among n tennis players is given by the polynomial , find the number of singles tennis matches that can be played among four tennis players.

Homework•Start finding articles for your quarter project.

•Chapter P Review Exercises:• Number 25 – 81, odds