SEMA 102 - College and Advanced Algebra Lesson 1 - Basic Concepts of Algebra PDF

SEMA 102 – College and Advanced Algebra Lesson 1. BASIC CONCEPTS OF ALGEBRA Overview: Algebra is a powerful mathematical tool that is used to solve real-word problems in science, business, and many other fields. This lesson reviews fundamental concept...

SEMA 102 – College and Advanced Algebra Lesson 1. BASIC CONCEPTS OF ALGEBRA Overview: Algebra is a powerful mathematical tool that is used to solve real-word problems in science, business, and many other fields. This lesson reviews fundamental concepts of algebra that are prerequisites for the study of higher Mathematics. Throughout the module, you will see how the special language of algebra describes your world. Lesson Objectives: After successful completion of this lesson, you should be able to: a. identify various kind and properties of real numbers, b. appreciate the real number system and its properties, c. use interval notation to write set of numbers, d. use properties of exponents in simplifying exponential expressions, e. use scientific notation, f. identify the terms, coefficients, and degree of polynomial; g. add, subtract, and multiply polynomials; h. factor polynomials using different methods or rules, i. determine the domain and range of expressions, j. simplify rational and radical expressions, and k. perform computations involving complex numbers. 1.1 The Real Number System Several sets of numbers are used extensively in algebra. The numbers that you are familiar with in day- today calculations are elements of the set of real numbers. Table 1.1. Important Subsets of the Real Numbers The set of real numbers is formed by taking the union of the sets of rational numbers and irrational numbers. Thus, every real number is either rational or irrational, as shown in Figure 1. The symbol is used to represent the set of real numbers. Figure 1.1. The Real Number System The Real Number Line The real number line is a graph used to represent the set of real numbers. An arbitrary point, called the origin, is labeled 0. Select a point to the right of 0 and label it 1. The distance from 0 to 1 is called the unit distance. Numbers to the right of the origin are positive and numbers to the left of the origin are negative. The real number line is shown in Figure 1.2. Figure 1.2 The Real Number Line On the real number line, the real numbers increase from left to right. The lesser of two real numbers is the one farther to the left on a number line. The greater of two real numbers is the one farther to the right on a number line. Absolute Value The absolute value of a real number 𝑎, denoted by 𝑎 , is the distance from 0 to 𝑎 on the number line. This distance is always taken to be nonnegative. The absolute value of −3 is 3 because −3 is 3 units from 0 on the number line. The absolute value of 5 is 5 because 5 is 5 units from 0 on the number line. The absolute value of a positive real number or 0 is the number itself. The absolute value of a negative real number, such as −3 , is the number without the negative sign. Absolute value is used to find the distance between two points on a real number line. If a and b are any real numbers, the distance between a and b is the absolute value of their difference. 1.2 Interval Notation The set {𝑥  𝑥 ≥ 3} represents all real numbers greater than or equal to 3. This set can be illustrated graphically on the real number line. By convention, a closed circle or a square bracket [ is used to indicate that an “endpoint” (x = 3) is included in the set. This interval is a closed interval because its endpoint is included.  The set {𝑥  𝑥 > 3} represents all real numbers greater than or equal to 3. This set can be illustrated graphically on the number line. By convention, an open circle ◯ or a parentheses ( is used to indicate that an “endpoint” (x = 3) is not included in the set. This interval is an open interval because its endpoint is not included. Notice that the sets {𝑥  𝑥 ≥ 3} and {𝑥  𝑥 > 3} consist of an infinite number of elements that cannot all be listed. Another method to represent the elements of such sets is by using interval notation. To understand interval notation, first consider the real number line, which extends infinitely far to the left and right. The symbol ∞ is used to represent infinity. The symbol −∞ is used to represent negative infinity. To express a set of real numbers in interval notation, sketch the graph first, using the symbols ( ) or [ ]. Then use these symbols at the endpoints to define the interval. Using Interval Notation The endpoints used in interval notation are always written from left to right. That is, the smaller number is written first, followed by a comma, followed by the larger number. Parentheses ) or ( indicate that an endpoint is excluded from the set. Square brackets ] or [ indicate that an endpoint is included in the set. Parentheses are always used with −∞ and ∞. Table 1.2 lists nine possible types of intervals used to describe subsets of real numbers. Table 1.2. Intervals on the Real Number Line Examples: Using Interval Notation 1. Graph the sets on the number line, and express the set in interval notation. 3 a. {𝑥  𝑥 ≥ 3} b. {𝑥  𝑥 > 3} c. {𝑥  𝑥 ≤ }2 Solution: Set-Builder Notation Graph Interval Notation a. {𝑥  𝑥 ≥ 3} b. {𝑥  𝑥 > 3} 3 c. {𝑥  𝑥 ≤ 2} 2. Express each interval in set-builder notation and graph: a. (−1, 4] b. [2.5, 4] c. (−4, ∞). Solution: Interval Notation Set-Builder Notation Graph a. (−1, 4] b. [2.5, 4] c. (−4, ∞) 1.3 Algebraic Expressions Algebra uses letters, such as x and y, to represent numbers. If a letter is used to represent various numbers, it is called a variable. A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots, is called an algebraic expression. Here are some examples of algebraic expressions: Properties of Real Numbers and Algebraic Expressions When you use your calculator to add two real numbers, you can enter them in any order. The fact that two real numbers can be added in any order is called the commutative property of addition. You probably use this property, as well as other properties of re al numbers listed in Table 1.3, without giving it much thought. The properties of the real numbers are especially useful when working with algebraic expressions. For each property listed in Table 1.3, a, b, and c represent real numbers, variables, or algebraic expressions.  Table 1.3. Properties Real Numbers Other properties of real numbers: ❖ Closure Property a + b is a real number; when you add two real numbers, the result is also a real number Example: 2 and 3 are both real numbers; 2 + 3 = 5 and the sum, 5, is also a real number a - b is a real number; when you subtract two real numbers, the result is also a real number Example: 7 and 13; are both real numbers; 7 − 13 = −6 and the difference, −6, is also a real number (a)(b) is a real number; when you multiply two real numbers, the result is also a real number Example: 8 and −10; are both real numbers; (8)( −10) = −80 and the product, − 80, is also a real number 𝒂 𝒃 is a real number where b ≠ 0; when you divide two real numbers, the result is also a real number −20 Example: −20 and 5; are both real numbers; = −4 and the quotient, −4, is also a real number 5  ❖ Identity Property of Addition – For any real number a, 𝑎 + 0 = a. Example: 5 + 0 = 5 ❖ Zero Product Property – any real number multiplied by zero, the product is zero. a 0=0 Example: 15 0 = 0 1.4 Evaluating Algebraic Expressions Evaluating an algebraic expression means to find the value of the expression for a given value of the variable. Many algebraic expressions involve more than one operation. Evaluating an algebraic expression without a calculator involves carefully applying the following order of operations agreement: The Order of Operations Agreement 1. Perform operations within the innermost parentheses and work outward. 2. If the algebraic expression involves a fraction, treat the numerator and the 3. denominator as if they were each enclosed in parentheses. 4. Evaluate all exponential expressions. 5. Perform multiplications and divisions as they occur, working from left to right. 6. Perform additions and subtractions as they occur, working from left to right. Examples: 1. (16 – 2) + (16 ÷ 8)2 ) + 32 2. 2{82 – 7[32 − 4(32 + 1)](−1)} 3. Evaluate 7 + 5(𝑥 − 4)3 for 𝑥 = 6. Solution: Solution: Solution: 1.5 Exponential Expressions Many algebraic expressions involve exponents.  Properties of Integer Exponents Simplifying Exponential Expressions Properties of exponents are used to simplify exponential expressions. An exponential expression is simplified when No parentheses appear. No powers are raised to powers. Each base occurs only once. No negative or zero exponents appear. Examples: Simplifying Exponential Expressions Simplify: a. (−3𝑥 4 𝑦 5 )3 b. (−7𝑥𝑦 4 )(−2𝑥 5 𝑦 6 ) Solution: Solution: −3 −35𝑥 5𝑦4 4𝑥 2 c. d. ( ). 5𝑥 6𝑦 −8 𝑦 Solution: Solution:  Simplifying exponential expressions seems to involve lots of steps. There are common errors you should try to avoid along the way. The table at the left shows a list of common errors with correct simplification as well as the description of error. Scientific Notations Scientific notation was devised as a shortcut method of expressing very large and very small numbers. The principle behind scientific notation is to use a power of 10 to express the magnitude of the number. It is customary to use the multiplication symbol, × , rather than a dot, when writing a number in scientific notation. Properties of exponents are used to perform computations with numbers that are expressed in scientific notation. Examples: Computations with Scientific Notation 1.8 × 104 a. (6.1 × 105 )(4 × 10−9 ) b. 3 × 10−2 Solution: Solution: 1.6 Simplifying Algebraic Expressions The terms of an algebraic expression are those parts that are separated by addition. For example, consider the algebraic expression 8𝑥 − 10𝑦 + 𝑧 − 5, which can be expressed as 8𝑥 + (−10𝑦) + 𝑧 + (−5). This expression contains four terms, namely, 8𝑥, − 10𝑦, 𝑧, and − 5. The numerical part of a term is called its coefficient. If a term containing one or more variables is written without a coefficient, the coefficient is understood to be 1. Like terms are terms that have exactly the same variable factors. An algebraic expression is simplified when parentheses have been removed and like terms have been combined. Examples: Simplifying Algebraic Expressions 1. 5(3𝑥2 + 7𝑥) + 9(2𝑥2 + 4𝑥) 2. 4𝑥 + 2[7 − (𝑥 − 4)] 3. 7 − 4[3 − (4𝑦 − 5)]