College Algebra Introduction P1 The Real Number System

College Algebra Introduction P1 The Real Number System
P2 Integer and Rational Number Exponents P3 Polynomials

Introduction Welcome! Addendum Quarter Project Wikispaces website:
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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. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑥 3 − 𝑦 3 𝑥 2 +𝑥𝑦+ 𝑦 2 when x = 2 and y = -3

P1 - Evaluate You try! Evaluate 𝑥+2𝑦 2−4𝑧 when x = 3, y = -2 and z = -4

P1 – Properties of Addition
Closure 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 Multiplication
Closure ab is a unique number Commutative ab = ba Associative (ab)c = a(bc) Identity a·1 = 1·a = a Inverse 𝑎∙ 1 𝑎 = 1 𝑎 ∙𝑎=1

P1 – Property Identification
Which property do each of the following use? 2𝑎 𝑏=2 𝑎𝑏

Which property do each of the following use? 4 𝑥+3 =4𝑥+12 𝑎+5𝑏 +7𝑐= 5𝑏+𝑎 +7𝑐

Which property do each of the following use? 1 2 ∙2 𝑎=1∙𝑎 1∙𝑎=𝑎

We use properties to simplify: 6𝑥 2 First we will use the Commutative Property Then we will use the Associative Property

We use properties to simplify: 3 4𝑝+5 We will use the Distributive Property

Use properties to simplify: 5+3(2𝑥−6)

Use properties to simplify: 4𝑥−2[7−5 2𝑥−3 ]

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.

Identify which property of equality each equation has: 𝐼𝑓 3𝑎+𝑏=𝑐, 𝑡ℎ𝑒𝑛 𝑐=3𝑎+𝑏 5 𝑥+𝑦 =5(𝑥+𝑦)

Identify which property of equality each equation has: 𝐼𝑓 4𝑎−1=7𝑏𝑎𝑛𝑑 7𝑏=5𝑐+2, 𝑡ℎ𝑒𝑛 4𝑎−1=5𝑐+2 𝐼𝑓 𝑎=5 𝑎𝑛𝑑 𝑏 𝑎+𝑐 =72, 𝑡ℎ𝑒𝑛 𝑏 5+𝑐 =72

P1 Time for a break!

P2 – Integer Exponents Remember…. 𝑏 𝑛 =𝑏∙𝑏∙𝑏∙⋯∙𝑏. Multiplied n times. 𝑏 0 =1 So, 3 0 =1, =1 Be careful…. −7 0 =−1

P2 – Integer Exponents If b ≠ 0 and n is a natural number, then 𝑏 −𝑛 = 1 𝑏 𝑛 and 1 𝑏 −𝑛 = 𝑏 𝑛 Examples: 3 −2 = 1 4 −3 =

P2 – Integer Exponents Examples: 5 −2 7 −1 = You try: −2 4 −3 2

P2 – Integer Exponents You try: −4 −3 −2 −5 −𝜋 0

P2 – Properties of Exponents
Product 𝑏 𝑚 ∙ 𝑏 𝑛 = 𝑏 𝑚+𝑛 Quotient 𝑏 𝑚 𝑏 𝑛 = 𝑏 𝑚−𝑛 where b ≠ 0 Power 𝑏 𝑚 𝑛 = 𝑏 𝑚∙𝑛 𝑎 𝑚 𝑏 𝑛 𝑝 = 𝑎 𝑚𝑝 𝑏 𝑛𝑝 𝑎 𝑚 𝑏 𝑛 𝑝 = 𝑎 𝑚𝑝 𝑏 𝑛𝑝 where b ≠ 0

Simplify: 𝑎 4 ∙𝑎∙ 𝑎 3 𝑥 4 𝑦 3 𝑥 𝑦 5 𝑧 2

Simplify: 𝑎 7 𝑏 𝑎 2 𝑏 5 𝑢𝑣 3 5

Simplify: 2 𝑥 5 5𝑦 4 3

You Try: 5𝑥 2 𝑦 −4𝑥 3 𝑦 5

You Try: 3𝑥 2 𝑦𝑧 −4 3

P2 – Scientific Notation
A number written in Scientific Notation has the form: 𝑎∙ 10 𝑛 Where n is an integer and 1≤𝑎≤10 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

For 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

3.5 × 10 5 = 2.51× 10 −8 =

Divide: 1.4× × 10 12

P2 – Rational Exponents and Radicals
If n is an even positive integer and b ≥ 0, then 𝑏 1 𝑛 is the nonnegative real number such that 𝑏 1 𝑛 𝑛 =𝑏 If n is an odd positive integer, then 𝑏 1 𝑛 is the real number such that 𝑏 1 𝑛 𝑛 =𝑏 =5 because 5 2 =25

Examples: =5 because 5 2 =25 − =−4 because −4 3 =− =4 because 4 2 =16 − = −( )=−4 because −4 2 =−16

Examples: However… ( −16) 1 2 is not a real number because (𝑥) 2 =−16 If n is an even positive integer and b < 0, then 𝑏 1 𝑛 is a complex number….we will get to that later…

For all positive integers m and n such that m/n is in simplest form, and fro all real numbers b for which 𝑏 1 𝑛 is a real number. 𝑏 𝑚 𝑛 = 𝑏 1 𝑛 𝑚 = 𝑏 𝑚 1 𝑛 Example:

𝑏 𝑚 𝑛 = 𝑏 1 𝑛 𝑚 = 𝑏 𝑚 1 𝑛 Example: 32 − −3 4

Simplify: 2 𝑥 1 3 𝑦 𝑥 3 𝑦

You Try: 𝑎 3 4 𝑏 𝑎 2 3 𝑏 3 4 3

P2 –Radicals Radicals are expressed by 𝑛 𝑏 , 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 𝑏 1 𝑛 is a real number, then 𝑛 𝑏 = 𝑏 1 𝑛 If the index equals 2, then the radical 2 𝑏 = 𝑏 also known as the principle square root of b.

P2 –Radicals For all positive integers n, all integers m and all real numbers b such that 𝑛 𝑏 is a real number, 𝑛 𝑏 𝑚 = 𝑛 𝑏 𝑚 = 𝑏 𝑚 𝑛 This helps us switch between exponential form and radical expressions 2𝑎𝑏 3 = 2𝑎𝑏 3 2

P2 –Radicals We can evaluate… 3 8 = =2 Try on our calculator!

P2 –Radicals If n is an even natural number and b is a real number, then 𝑛 𝑏 𝑛 = 𝑏 4 16𝑧 4 =2 𝑧 If n is an odd natural number and b is a real number, then 𝑛 𝑏 𝑛 =𝑏 5 32𝑎 5 =2𝑎

P2 –Radical Properties If n and m are natural numbers and a and b are positive real numbers, then… Product 𝑛 𝑎 ∙ 𝑛 𝑏 = 𝑛 𝑎𝑏 Quotient 𝑛 𝑎 𝑛 𝑏 = 𝑛 𝑎 𝑏 Index 𝑚 𝑛 𝑎 = 𝑚𝑛 𝑎

P2 –Radicals How do we know if our expression is in simplest form?
The radicand contains only powers less than the index. The index of the radical is as small as possible. The denominator has been rationalized. Such that no radicals occur in the denominator. No fractions occur under the radical sign.

P2 –Radicals Simplify: 4 32𝑥 3 𝑦 4

P2 –Radicals Simplify: 3 162𝑥 4 𝑦 6

P2 –Radicals Like radicals have the same radicand and the same index… 4 3𝑥 −9 3𝑥 2 3 𝑦 𝑦 2 − 3 𝑦 2

P2 –Radicals Simplify: 5𝑥 3 16𝑥 4 − 3 128𝑥 7

P2 –Radicals Multiply: 5 6 −

P2 –Radicals You Try: 3− 𝑥−7 2

P2 –Radicals To 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 –Radicals Example: 5 3 𝑎

P2 –Radicals You Try: 3 32𝑦 where y > 0

P2 –Radicals Example: −4 5

P2 –Radicals You Try: 2+4 𝑥 3−5 𝑥 , where x > 0

P2 - Radicals LET’S TAKE A BREAK!

P3 - Polynomials A 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 −12 𝑎 2 𝑏 𝑐 3

P3 - Polynomials A 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. 5𝑥 4 − 6𝑥 3 + 5𝑥 2 −7𝑥−8 A binomial is a polynomial with two terms. 3𝑥 4 −7 A trinomial is a polynomial with three terms. 3𝑥 2 +6𝑥−1 **We always write our polynomials in descending order according to the largest exponent…

P3 - Polynomials Example: Add 3𝑥 2 − 2𝑥 2 −6 + 4𝑥 2 −6𝑥−7

P3 - Polynomials Example: Multiply 2𝑥−5 𝑥 3 −4𝑥+2

P3 - Polynomials Example: Multiply using FOIL – For BINOMIALS ONLY 4𝑥+5 3𝑥−7

P3 - Polynomials Special Forms – 𝑥+𝑦 𝑥−𝑦 = 𝑥 2 − 𝑦 2 𝑥+𝑦 2 = 𝑥 2 +2𝑥𝑦+ 𝑦 2 𝑥−𝑦 2 = 𝑥 2 −2𝑥𝑦+ 𝑦 2 These are for your reference, you do not have to use the special form rules, you can simply multiply manually.

P3 - Polynomials Example: 7𝑥+10 7𝑥−10

P3 - Polynomials Example: Evaluate the polynomial 2𝑥 3 − 6𝑥 2 +7 for x = -4

P3 - Polynomials Example: The number of singles tennis matches that can be played among n tennis players is given by the polynomial 𝑛 2 − 1 2 𝑛, 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

College Algebra Introduction P1 The Real Number System

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