Chapter P Prerequisites: Fundamental Concepts of Algebra Pre-calculus notes Date: P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Algebraic expression: a combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots Examples: Exponential notation: If n is a counting number (1,2,3, etc.), b n = b is the and n is the Evaluating an algebraic expression: find the value of the expression for a given value of the variable Order of Operations 1. Start at the innermost set of parentheses and word outward. 2. Evaluate all exponential expressions. 3. Perform multiplication and division left to right. 4. Perform addition and subtraction left to right. Ex. 1: Evaluate 8 + 6(x 3) 2 for x = 6. When an equal sign is placed between two algebraic expressions, an is formed. A formula is an that uses variables to express a relationship between two or more quantities. Mathematical modeling: the process of finding formulas to describe real-world phenomena Ex. 2: If the average cost of tuition and fees, T, for public four-year colleges, adjusted for inflation is modeled by the formula T = 17x 2 + 261x + 3257 where x is the number of years since the end of the school year in 2000. Use the formula to project the average cost of tuition and fees at public U.S. colleges for the school year ending 2010.
Set: a collection of objects whose contents can be clearly determined The objects in a set are called the of the set. Ways to represent a set: Roster method (listing all the elements, separated by commas. Set-builder notation: Intersection of sets A and B: the set of elements common to both set A and set B Notation: Venn diagram: Ex. 3: Find the intersection: {3,4,5,6,7} {6,8,10,12} Union of sets A and B: the set of elements that are members of set A or of set B or of both sets. Notation: Venn diagram: Ex. 4: Find the union: {3,4,5,6,7} {3,7,8,9} A set with no elements is called or. Hmk: Math XL: Log in and that complete Day 1 assignment to practice using the features of the program. Also use the code to view the e-book and complete the following assignment: Pgs. 15-16: #3-48 (mult of 3) check odd answers make sure to write out the problems and show all of your work.
Chapter P Prerequisites: Fundamental Concepts of Algebra Pre-calculus notes Date: P.1: Algebraic Expressions, Mathematical Models, and Real Numbers (cont.) Subsets of the Real Numbers Name Symbol Description Examples Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real numbers: the set of numbers that are either rational or irrational. Ex. 5: Consider the following set of numbers: { 9, 1.3,0,0.3, π, 9, 10} 2 List the members in the set that are: Natural numbers Whole numbers Integers Rational numbers Irrational members Real numbers Ordering the real numbers Symbols : and
Absolute Value: a Informal def: Formal def: x = { Ex. 6: Rewrite without absolute value bars: a) 1 2 b) π 3 c) x x if x>0. Distance between Points on a Real Number Line: a and b are any two pts on a real number line, then the distance between a and b is given by: Ex. 7: Find the distance between -4 and 5. = Properties of Real Numbers and Algebraic Expressions Name Meaning Example Commutative Property of Addition/Multiplication Associative Property of Addition/Multiplication Distributive Property Identity Property of Addition Identity Property of Multiplication Inverse Property of Addition Inverse Property of Multiplication Definition of Subtraction: a b = -b is called the of b. Definition of Division: a b = 1 b is called the of b.
Simplifying Algebraic Expressions: Combine like terms (add their coefficients) Ex. 8: Simplify: 6(2x 2 + 4x) + 10(4x 2 + 3x) Properties of Negatives: Let a and b represent real numbers, variables, or algebraic expressions. Property (-1)a = Example -(-a) = (-a)b = a(-b) = -(a+b) = -(a-b) = If a negative sign or a subtraction symbol appears outside parentheses, drop the parentheses and change the sign of every term within the parentheses. (Distribute the negative.) Ex. 9: Simplify: 6 + 4[7 (x 2)] Ex. 10: Name the property illustrated by each statement. a) 11 (7 + 4) = 11 7 + 11 4 b) 7 (11 8) = (11 8) 7 c) 1 x+3 (x + 3) = 1, x 3 Pgs. 16-19: # 51-159 (mult of 3, skip 144, 147), 160-162 (all) check odd answers
P.3 Radicals and Radical Expressions Pre-calculus notes Date: Def: If b 2 = a, then b is a of a. Symbol: represents the positive or of a number. The symbol is called a and the number under the sign is called the. Together they form a. Ex. 1: Evaluate: a) 81 = b) 9 = c) 1 25 = d) 36 + 64 = e) 36 + 64 = Note: The square root of a negative number is NOT a real number. Def: For any real number a, a 2 = Product Rule for Square Roots: If a and b represent non-negative real numbers, then and. A square root is simplified when: Ex. 2: Simplify. a) 75 b) 5x 10x
Quotient Rule for Square Roots If a and b represent non-negative real numbers and b 0, then and. Ex. 3: Simplify. a) 25 150x3 = b) 16 2x Adding and Subtracting Square Roots: Two or more radicals can be added/subtracted only if they are. Ex. 4: Add or subtract as indicated. a) 8 13 + 9 13 = b) 17x 20 17x = Ex. 5: Add or subtract as indicated. It may be necessary to simplify first. a) 5 27 + 12 b) 6 18x 4 8x A radical expression is not simplified if there is a radical in the denominator. The process of rewriting the expression to eliminate any radicals in the denominator is called the. Ex. 6: Simplify. a) 5 3 b) 6 12 Def: a + b and a b are called. Multiply them together and the result is:
Ex. 7: Rationalize the denominator Use the conjugate! 8 4 + 5 Other kinds of roots: The principal nth root of a real number a, symbolized by is defined by the following: n is called the If n is even, then a and b are. If n is odd, then a and b. Cube Roots Fourth Roots Fifth Roots The product and quotient rules apply to all higher roots as well. Ex. 8: Simplify. 3 a) 40 5 b) 8 5 8 3 c) 125 27
Rational Exponents: n Def: If a represents a real number where n is an, then n a = Also, a 1 n = Ex. 10: Simplify. a) 25 1 2 = b) 8 1 3 = c) 81 1 4 = d) ( 8) 1 3 = e) 27 1 3 = n Def: If a m represents a real number and m n is a, where n, then a m n = Also, a m n = Ex. 11: Simplify. a) 27 4 3 b) 4 3 2 c) 32 2 5
Ex. 12: Simplify using properties of exponents. a) (2x 4 3) (5x 8 3) b) 20x4 5x 3 2 Rational exponents are sometimes useful for simplifying radicals by reducing the index. 6 Ex. 13: Simplify: x 3. Homework: Pgs. 46-48: #3-108 (mult of 3), 123, 125, 133, 137
P.4 Polynomials Pre-calculus notes Date: A polynomial is a single term or the sum of two or more terms containing variables with whole number exponents. The standard form of a polynomial is found by writing the terms in powers of the variable. If a 0, the degree of ax n is. The degree of a nonzero constant is. A polynomial with one term is called a. A polynomial with two terms is called a. A polynomial with three terms is called a. The degree of a polynomial is. Definition of a Polynomial in x A polynomial in x is an algebraic expression of the form Where are real numbers and a n 0 and n is a nonnegative integer. The polynomial is of degree, a n is the, and a 0 is the. Adding and Subtracting Polynomials Ex. 1 Perform the indicated operations and simplify. a) ( 17x 3 + 4x 2 11x 5) + (16x 3 3x 2 + 3x 15) b) (13x 3 9x 2 7x + 1) ( 7x 3 + 2x 2 5x + 9)
Multiplying Polynomials: Use properties of exponents. To multiply polynomials when neither is a monomial, multiply each term of one polynomial by each term of the other polynomial. Then combine like terms. Ex. 2: Multiply: (5x 2)(3x 2 5x + 4) To multiply two binomials, FOIL (First-Outer-Inner-Last) Ex. 3: Multiply: (7x 5)(4x 3) Special Products 1) Product of Sum and Difference of Two Terms (a + b)(a b) = 2) The Square of a Binomial Sum (a + b) 2 = 3) The Square of a Binomial Difference (a b) 2 = 4) The Cube of a Binomial Sum (a + b) 3 = 5) The Cube of Binomial Difference (a b) 3 =
Ex. 4: Multiply. a) (7x + 8)(7x 8) = b) (2y 3 5)(2y 3 + 5) = Ex. 5: Multiply. a) (x + 10) 2 = b) (5x + 4) 2 = Ex. 6: Multiply. a) (x 9) 2 = b) (7x 3) 2 = Polynomials in two variables x and y. The degree of ax n y m is.
Ex. 7: Subtract: (x 3 4x 2 y + 5xy 2 y 3 ) (x 3 6x 2 y + y 3 ). Ex. 8: Multiply. a) (7x 6y)(3x y) b) (2x + 4y) 2 Ex. 9: Perform the indicated operations. a) (3x + 4y) 2 (3x 4y) 2 b) (5x 3)6 (5x 3) 4 Homework: Pgs. 58-59: #3-81 (mult of 3), 91, 94, 107, 108 Quiz over P.1-P.4 this Thursday 9/1
P.5 Factoring Polynomials Pre-calculus notes Date: Factoring a polynomial containing the sum of monomials means finding an equivalent expression that is a product. We will be factoring over the set of integers. All coefficients will be integers. Polynomials that cannot be factored using integer coefficients are called over the integers or. The first step in any factoring problem is to look for the. Ex. 1: Factor. a) 10x 3 4x 2 b) 2x(x 7) + 3(x 7) When factoring a polynomial with four terms it may be possible to factor by grouping. Ex. 2: Factor: x 3 + 5x 2 2x 10. Factoring trinomials of the form ax 2 + bx + c Find two First terms whose product is ax 2. Find two Last terms whose product is c. By trial and error try all arrangements of the factors until the sum of the Outside product and the Inside product is bx. If no such combination exists, the polynomial is. Ex. 3: Factor: x 2 + 13x + 40. Ex. 4: Factor: x 2 5x 14. Ex. 5: Factor: 6x 2 + 19x 7.
Ex. 6: Factor: 3x 2 13xy + 4y 2. Factoring the Difference of Two Squares: a 2 b 2 = Ex. 7: Factor: a) x 2 81 b) 36x 2 25 The goal of factoring is to factor COMPLETELY. Ex. 8: Factor (completely): x 4 81. Factoring Perfect Square Trinomials a 2 + 2ab + b 2 = a 2 2ab + b 2 = In a perfect square trinomial: The first and last terms are squares of monomials or integers. The middle term is twice the product of the expressions being squared in the first and last terms. Ex. 9: Factor. a) x 2 + 6x + 9 b) 25x 2 60x + 36 Factoring Sums and Differences of Cubes a 3 + b 3 = a 3 b 3 =
Ex. 10: Factor. a) x 3 + 1 b) 125x 2 8 Refer to the strategy for factoring polynomials on Pg. 69 in the textbook. Ex. 11: Factor: 3x 3 30x 2 + 75x Start with the GCF! Factoring by grouping can include grouping a trinomial. Ex. 12: Factor: x 2 36a 2 + 20x + 100. Factoring with fractional and negative exponents. Recall: Expressions containing fractional exponents are NOT polynomials. They can be simplified using factoring techniques. Ex. 13: Factor and simplify: x(x 1) 1 2 + (x 1) 1 2. Quiz Thursday over sections P.1-P.4. Homework: Math XL assignment: P.1-P.4 review (due by class time) Due Friday (sect. P-5): Pgs. 71-73: #3-99 (mult of 3), 134-137 (all) *You may work on the assignment due Friday after you finish your quiz. I will be checking for all of the supplies (3-ring binder, graph paper, calculator) during the quiz on Friday.
P.6 Rational Expressions Pre-calculus notes Date: A rational expression is the of two polynomials. The set of real numbers for which an algebraic expression is defined is the of the expression. Because division by zero is undefined, all values that make the denominator zero must be excluded. Ex. 1: Find all the numbers that must be excluded from the domain of each rational expression: a) 7 x + 5 b) x x 2 36 A rational expression is simplified if its numerator and denominator have no common factors other than 1 and -1. To simplify a rational expression: Factor the numerator and the denominator completely. Divide both the numerator and denominator by any common factors. Ex. 2: Simplify: a) x 3 +3x 2 x+3 b) x 2 1 x 2 +2x+1 To multiply rational expressions: Factor all numerators and denominators completely. Divide numerators and denominators by common factors. Multiply the remaining factors in the numerators and multiply the remaining factors in the denominators.
Ex. 3: Multiply: x+3 x 2 4 x2 x 6 x 2 +6x+9. To divide two rational expressions, the problem must be rewritten as a multiplication problem by using the reciprocal Ex. 4: Divide: x2 2x+1 x2 +x 2 x 3 +x 3x 2 +3 To add and subtract rational expressions they must have common denominators. Be careful to use the distributive property correctly. Ex. 5: Subtract: x 3x+2 x+1 x+1 Ex. 6: Add: 3 x+1 + 5 x 1
To find the least common denominator, or LCD: Factor each denominator completely. List all the factors of the first denominator. Add to the list in set 2 any factors of the second denominator that do not appear in the list. Find the product of the factors in Step 3. Ex. 7 Find the LCD of the following rational expressions: 3 x 2 6x+9 and 7 x 2 9 Ex. 8: Subtract: x x 4. x 2 10x+25 2x 10 have numerators and denominators containing one or more rational expressions. (A fraction within a fraction.) There are two methods for simplifying. Method #1: Combine the numerator into a single expression and combine the denominator into a single expression. Then perform the division by rewriting as a multiplication problem. Ex. 9: Simplify using Method #1. 1 x 3 2 1 x +3 4.
Method #2: Find the least common denominator of all the rational expressions in the numerator and denominator. Then multiply each term in the numerator and denominator by this LCD. (Preferred method.) Ex. 10: Simplify using Method #2. 1 x+7 1 x 7. Homework due Tuesday: Pgs. 82-83: #3-69 (mult of 3) Homework due Wednesday: Pg. 88: #1-32 Homework due Thursday by class time: Math XL: Ch. P Review Ch. P Test on Thursday 9/8 Homework due Friday by class time: Math XL: Ch. 1 Preview