Elementary Algebra Exam Material Familiar Sets of Numbers • Natural numbers – Numbers used in counting: 1, 2, 3, … (Does not include zero) • Whole numbers – Includes zero and all natural numbers: 0, 1, 2, 3, … (Does not include negative numbers) • Fractions – Ratios of whole numbers where bottom number can not be zero: , , , etc Top number is called " numerator" Bottom number is called " denominator" Prime Numbers • Natural Numbers, not including 1, whose only factors are themselves and 2, 3, 5, 7, 11, 13, 17, 19, 23, etc • What is the next biggest prime number? 29 Composite Numbers • Natural Numbers, bigger than 1, that are not prime 4, 6, 8, 9, 10, 12, 14, 15, 16, etc • Composite numbers can always be “factored” as a product (multiplication) of prime numbers Factoring Numbers • To factor a number is to write it as a product of two or more other numbers, each of which is called a factor 12 = (3)(4) & are factors 12 = (6)(2) & are factors 12 = (12)(1) 12 and are factors 12 = (2)(2)(3) 2, 2, and are factors In the last case we say the 12 is “completely factored” because all the factors are prime numbers Hints for Factoring Numbers • • • • To factor a number we can get two factors by writing any multiplication problem that comes to mind that is equal to the given number Any factor that is not prime can then be written as a product of two other factors This process continues until all factors are prime Completely factor 28 28 = (4)(7) 28 = (2)(2)(7) & are factors, but is not prime is written as (2)(2), both prime In the last case we say the 28 is “completely factored” because all the factors are prime numbers Other Hints for Factoring • • • • Some people prefer to begin factoring by thinking of the smallest prime number that evenly divides the given number If the second factor is not prime, they again think of the smallest prime number that evenly divides it This process continues until all factors are prime Completely factor 120 120 = (2)(60) 60 is not prime, and is divisible by 120 = (2)(2)(30) 30 is not prime, and is divisible by 120 = (2)(2)(2)(15) 30 is not prime, and is divisible by 120 = (2)(2)(2)(3)(5) all factors are prime In the last case we say the 120 is “completely factored” because all the factors are prime numbers Fundamental Principle of Fractions • If the numerator and denominator of a fraction contain a common factor, that factor may be divided out to reduce the fraction to lowest terms: • Reduce to lowest terms by factoring: 12 18 1 12 ⋅ ⋅ = = 18 21 ⋅ ⋅ 31 When common factors are divided out, "1" is left in each place Summarizing the Process of Reducing Fractions • • Completely factor both numerator and denominator Apply the fundamental principle of fractions: divide out common factors that are found in both the numerator and the denominator When to Reduce Fractions to Lowest Terms • Unless there is a specific reason not to reduce, fractions should always be reduced to lowest terms • A little later we will see that, when adding or subtracting fractions, it may be more important to have fractions with a common denominator than to have fractions in lowest terms Homework Problems • • • • • Section: 1.7 Page: 74 Problems: All: – 30, 35 – 50, 55 – 80 MyMathLab Homework 1.7 for practice MyMathLab Homework Quiz 1.7 is due for a grade on the date of our next class meeting Terminology of Algebra • Constant – A specific number Examples of constants: • −6 Variable – A letter or other symbol used to represent a number whose value varies or is unknown Examples of variables: x n A Terminology of Algebra • Expression – constants and/or variables combined with one or more math operation symbols for addition, subtraction, multiplication, division, exponents and roots in a meaningful way Examples of expressions: • 2+3 5+ x 10 4− n y −9⋅w Only the first of these expressions can be simplified, because we don’t know the numbers represented by the variables Terminology of Algebra • Term – an expression that involves only a single constant, a single variable, or a product (multiplication) of a constant and variables Examples of terms: • A⋅ B ⋅y AB y Note: When constants and variables are multiplied, or when two variables are m − 5⋅ x multiplied, it is common to omit the multiplication symbol Previous example is commonly written: m − 5x Terminology of Algebra • • Every term has a “coefficient” Coefficient – the constant factor of a term – (If no constant is seen, it is assumed to be 1) • What is the coefficient of each of the following terms? 2 − 5x m AB −5 y 3 Like Terms • • • constant _ product variable constant Like Terms: terms are called “like terms” if they have exactly the same variables variables with exactly the same exponents, but may have different coefficients Recall that a term is a _ , a , or a _ of a and Example of Like Terms: x y and − 7x y Determine Like Terms • • − 4xy Given the term: Which of the following are like terms? 5x y 3 xy − 2x y 54 xy Adding Like Terms • When “like terms” are added, the result is a like term and its coefficient is the sum of the coefficients of the other terms • • Example: The reason for this can be shown by the distributive property: 2x + 7x = 9x x + x = ( + 7) x = x Subtracting Like Terms • When like terms are subtracted, the result is a like term with coefficient equal to the difference of the coefficients of the other terms • • Example: Reasoning: x − x = − 5x x − x = ( − ) x = −5 x Simplifying Expressions by Combining Like Terms • Any expression containing more than one term may contain like terms, if it does, all like terms can be combined into a single like term by adding or subtracting as indicated by the sign in front of each term • Example: Simplify: Middle two steps can be done in your head! x − 19 y + x + y − x x + x − x − 19 y + y ( + − 1) x + ( − 19 + 2) y x − 17 y Review of Distributive Property • Distributive Property – multiplication can be distributed over addition or subtraction • Some people make the mistake of trying to distribute multiplication over multiplication • Example: 3( x + y ) = x + y • 3( xy ) = xy ≠ x3 y Associative Property justifies answer! !! + or – in Front of Parentheses • When a + or – is found in front of a parentheses, we assume that it means “positive one” or “negative one” • Examples: − ( + y ) = − 1( + y ) = − − y = −1 − y + ( x − ) − ( x − ) = + 1( x − ) − 1( x − 4) = 3+ x − 2− x + = Multiplying Terms • Terms can be combined into a single term by addition or subtraction only if they are like terms • Terms can always be multiplied to form a single term by using commutative and associative properties of multiplication • Example: xy + x Won' t simplify! ( xy ) ( 3x ) = ( 2)( 3) x( x ) y = 6x y Middle step can be done in your head! Simplifying an Expression • • • Get rid of parentheses by multiplying or distributing Combine like terms Example: − 3( − x ) + 5( x + ) + 2( − x ) + − x x + x + 10 − x + − x x + 14 Homework Problems • • • Section: 1.8 Page: 80 Problems: All: – 30 Odd: • • 33 – 75 MyMathLab Homework 1.8 for practice MyMathLab Homework Quiz 1.8 is due for a grade on the date of our next class meeting