3 P1 The Real Number System Bonus opportunity for the beginning of P1 on the Wikispaces site!Number systemPrime/Composite NumbersAbsolute ValueExponential NotationOrder of Operations
4 P1 - EvaluateTo 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
5 P1 - EvaluateYou try! Evaluate 𝑥+2𝑦 2−4𝑧 when x = 3, y = -2 and z = -4
6 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
7 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
8 P1 – Property Identification Which property do each of the following use?2𝑎 𝑏=2 𝑎𝑏
9 P1 – Property Identification Which property do each of the following use?4 𝑥+3 =4𝑥+12𝑎+5𝑏 +7𝑐= 5𝑏+𝑎 +7𝑐
10 P1 – Property Identification Which property do each of the following use?1 2 ∙2 𝑎=1∙𝑎1∙𝑎=𝑎
11 P1 – Property Identification We use properties to simplify: 6𝑥 2 First we will use the Commutative Property Then we will use the Associative Property
12 P1 – Property Identification We use properties to simplify: 3 4𝑝+5 We will use the Distributive Property
13 P1 – Property Identification Use properties to simplify: 5+3(2𝑥−6)
14 P1 – Property Identification Use properties to simplify: 4𝑥−2[7−5 2𝑥−3 ]
15 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.
16 P1 – Properties of Equality Identify which property of equality each equation has: 𝐼𝑓 3𝑎+𝑏=𝑐, 𝑡ℎ𝑒𝑛 𝑐=3𝑎+𝑏 5 𝑥+𝑦 =5(𝑥+𝑦)
17 P1 – Properties of Equality 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
27 P2 – Properties of Exponents You Try: 5𝑥 2 𝑦 −4𝑥 3 𝑦 5
28 P2 – Properties of Exponents You Try: 3𝑥 2 𝑦𝑧 −4 3
29 P2 – Scientific Notation A number written in Scientific Notation has the form:𝑎∙ 10 𝑛Where n is an integer and 1≤𝑎≤10For 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 moved7, 430, 000
30 P2 – Scientific Notation 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
33 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
34 P2 – Rational Exponents and Radicals Examples: =5 because 5 2 =25 − =−4 because −4 3 =− =4 because 4 2 =16 − = −( )=−4 because −4 2 =−16
35 P2 – Rational Exponents and Radicals 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…
36 P2 – Rational Exponents and Radicals 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:
39 P2 – Rational Exponents and Radicals You Try: 𝑎 3 4 𝑏 𝑎 2 3 𝑏 3 4 3
40 P2 –RadicalsRadicals 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.
41 P2 –RadicalsFor 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
42 P2 –RadicalsWe can evaluate… 3 8 = =2 Try on our calculator!
43 P2 –RadicalsIf 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
44 P2 –Radical PropertiesIf 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
45 P2 –Radical PropertiesIf 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
46 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.
53 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…
59 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 −12 𝑎 2 𝑏 𝑐 3
60 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. 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…
63 P3 - PolynomialsExample: Multiply using FOIL – For BINOMIALS ONLY 4𝑥+5 3𝑥−7
64 P3 - PolynomialsSpecial 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.
66 P3 - PolynomialsExample: Evaluate the polynomial 2𝑥 3 − 6𝑥 2 +7 for x = -4
67 P3 - PolynomialsExample: 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.
68 Homework Start finding articles for your quarter project. Chapter P Review Exercises:Number 25 – 81, odds