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Transcript
COLLEGE ALGEBRA CHAPTER R REAL NUMBERS • In applications of algebraic concepts, we use real numbers to represent quantities such as distance, time, speed, area, profit, loss, and temperature. • Some frequently used set of numbers and relationships among them are shown below. RATIONAL NUMBERS • Numbers that can be expressed in the form p/q, where p and q are integers and q does not equal zero are rational numbers. • Decimal notation for rational numbers either terminates (ends) or repeats. • Each of the following is a rational number: • • • • 0 -7 ¼ = .25 ----5/11 = -0.45454545…. (repeating) = -0.45 IRRATIONAL NUMBERS • The real numbers that are not rational are irrational. • Decimal notation for irrational numbers neither terminates or repeats. • Each of the following is an irrational number • 𝜋 • 2 • -6.12122122212222….. • Although this last irrational shows a pattern, it is NOT repeating a block of digits REAL NUMBERS • Rational and irrational numbers create the set of real numbers • Real numbers can be modeled in a number line • Each point on the line represents a real number, and every real number is represented by a point on the line • The order of the real numbers can be determined from the number line. • If a number a is to the left of a number b, then a is less than b (a < b) • Similarly, a is greater than b ( a > b) if a is to the right of b on the number line • The statement a < b, read “a is less than or equal to b”, is true if either a < b or a = b is true. ELEMENT • The symbol is used to indicate that a member, or element, belongs to a set. • Thus if we let ℚ represent the set of rational numbers, then – 7 ℚ • We can also write 2 ∉ ℚ • This indicates that the square root of 2 is not a rational number. • When all the elements of one set are elements of a second set, we say that the first set is a subset of the second set. • The symbol used to denote this is ⊆ • The rationals are a subset of the reals: ℚ ⊆ ℝ INTERVAL NOTATION • Sets of real numbers can be expressed using interval notation. • The real numbers a and b such that a < b, create the open interval (a,b) as the set of real numbers between but not including a and b • (a,b) = 𝑥 𝑎 < 𝑥 < 𝑏 • The points a and b are endpoints of the interval. The parentheses indicate that the endpoints are not included. • Some intervals extend without bound in one or both directions. The interval [a,∞) begins at a and extends to the right without bound • [a,∞) = 𝑥 𝑥 ≥ 𝑎 • The bracket indicates that a is included in the interval INTERVALS: TYPES, NOTATION, GRAPHS Open Half Open Half Open Closed PROPERTIES OF REAL NUMBERS ABSOLUTE VALUE • The number line can be used to provide a geometric interpretation of absolute value • The absolute value of a number a, denoted by 𝑎 is its distance from 0 on the number line. • Absolute value can be used to find the distance between any two points on the number line as well. ABSOLUTE VALUE • For any real number a, 𝑎, 𝑎 = −𝑎, 𝑖𝑓 𝑎 ≥ 0 𝑖𝑓 𝑎 < 0 When a is nonnegative, the absolute value of a is a. When a is negative, the absolute value of a is the opposite, or additive inverse of a. Thus the absolute value of a is nonnegative for any real number. • For any real numbers a and b, the distance between a and b is 𝑎 − 𝑏 , or equivalently 𝑏 − 𝑎 INTEGERS AS EXPONENTS • For any positive integer n, 𝑎𝑛 = 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ ∙ ∙ 𝑎, (n factors) where a is the base and n is the exponent. • For any nonzero real number a and integer m, 𝑎0 = 1 And 𝑎−𝑚 = 1 𝑎𝑚 SIMPLIFY • For any nonzero numbers a and b and any integer m and n 𝑎−𝑚 𝑏𝑛 = 𝑚 −𝑛 𝑏 𝑎 • A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed PROPERTIES OF EXPONENTS • For any real numbers a and b and any integers m and n, assuming 0 is not raised to a nonpositive power: • Product Rule: 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛 • 𝑎𝑚 Quotient Rule: 𝑛 = 𝑎𝑚−𝑛 𝑎 Power Rule: (𝑎𝑚 )𝑛 =𝑎𝑚𝑛 (a ≠ 0) • • Raising a product to a power: (𝑎𝑏)𝑚 = 𝑎𝑚 𝑏𝑚 • Raising a quotient to a power: 𝑎 𝑚 𝑎𝑚 ( ) = 𝑚 𝑏 𝑏 (𝑏 ≠ 0) RULES FOR ORDER OF OPERATIONS • Do all calculations within grouping symbols before operations outside. When nesting grouping symbols are present, work from the inside out. • Evaluate all exponential expressions • Do all multiplication and division in order from left to right • Do all addition and subtraction in order from left to right • PEMDAS SCIENTIFIC NOTATION • Scientific notation for a number is an expression of the type N x 10𝑚 Where 1 < N < 10 N is in decimal notation And M is an integer PROBLEM SETS R. 1 R. 2 #4-76 #4-76 4s 4s R.3 •Addition, Subtraction, and Multiplication of Polynomials POLYNOMIALS • Polynomials in One Variable • A polynomial in one variable is any expression of the type 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 1 + 𝑎0 Where n is a nonnegative integer and 𝑎𝑛 , …, 𝑎0 are real numbers, called coefficients. The parts of a polynomial separated by plus signs are called terms. The leading coefficient is 𝑎𝑛 and the constant term is 𝑎𝑜 . If 𝑎𝑛 ≠ 0, the degree of the polynomial is n. The polynomial is said to be written in descending order because the exponents decrease from left to right. VOCAB • An algebraic expression like 3ab³ - 8 is a polynomial of several variables. • The degree of a term is the sum of the exponents of the variables in that term. • The degree of a polynomial is the degree of the term of the highest degree • A monomial is a polynomial with one term • A binomial is a polynomial with two terms • A trinomial is a polynomial with three terms ADDITION AND SUBTRACTION • If two terms of an expression have the same variables raised to the same powers, they are called like terms. • Also sometimes referred to as similar terms • We can combine or collect like terms using the distributive property. • For example: • 3y² + 5y² = (3+5)y² = 8y² MULTIPLICATION • Multiplication of polynomials is based on the distributive property • For example: • (x+4)(x+3) = x(x+3) + 4(x+3) = x² + 3x + 4x + 12 = x² + 7x + 12 • Some of you may have learned this process as FOIL or lattice – both of those are other accepted methods to multiply polynomials. x +3 x x² 3x +4 4x 12 SPECIAL PRODUCTS • These are some of the special products that come as a result of multiplying two binomials. • (a + b)² = a² + 2ab + b² • (a - b)² = a² - 2ab + b² • (a + b)(a – b) = a² - b² • These are some of the special products that come as a result of multiplying a binomial and a trinomial • a³ + b³ = (a + b)(a² - ab + b²) • a³ - b³ = (a - b)(a² + ab + b²) R.4 FACTORING TERMS WITH COMMON FACTORS • When a polynomial is to be factored, we should always look first to factor out a factor that is common to all the terms using the distributive property. We generally look for the constant common factor with the largest absolute value and for variables with the largest exponent common to all the terms. In this sense, we factor out the largest common factor. FACTORING EXAMPLE • Factor the following • 15 – 10x – 5x² • We can see that the largest common factor here is 5. • Thus: 5(3 - 2x - x²) = 15 – 10x – 5x² • 12x²y² - 20x³y • We can see that the largest common factor here is 4x²y • Thus: 4x²y (3y – 5x) = 12x²y² - 20x³y FACTOR BY GROUPING • In some polynomials, pairs of terms have a common binomial factor that can be removed through a process called factor by grouping. • Example: Factor x³ + 3x² - 9x – 27 • The grouping that we should consider here – • (x³ + 3x²) + (-9x - 27) • x²(x+3) – 9(x + 3) • (x² - 9) (x+3) • NOT DONE YET • (x – 3)(x + 3)(x + 3) or (x – 3)(x + 3)² TRINOMIALS OF THE TYPE X²+BX+C • Some trinomials can be factored into the product of two binomials. To factor a trinomial of the form x² + bx + c, we look for binomial factors of the form (x + p)(x + q) where p times q = c and p plus q =b. That is, we look for two numbers p and q whose sum is the coefficient of the middle term of the polynomial, b and whose product is the constant term, c. PROBLEM SETS R. 3 R. 4 #2-52 Evens #6-120 6s RATIONAL EXPRESSIONS • A rational expression is the quotient of two polynomials. • The domain of an algebraic expression is the set of all real numbers for which the expression is defined • Since division by zero is not defined, any number that makes the denominator zero is not in the domain of the rational expression. EXAMPLES • Find the domain of 2 / (x – 3) • Find the domain of (x² - 4) / (x² - 5x + 6) SIMPLIFYING • To simplify a rational expression, we use the fact that 𝑎∙𝑐 𝑎 𝑐 𝑎 𝑎 = ∙ = ∙1= 𝑏∙𝑐 𝑏 𝑐 𝑏 𝑏 • Simplify: 9𝑥 2 +6𝑥−3 12𝑥 2 −12 DEFINITIONS • Equivalent expressions • Two expressions have the same value for all numbers that are in both domains • Example 2−𝑥 −1 = 𝑥2 + 𝑥 − 6 𝑥 + 3 • Note that -3 is not in the domain of either expression, whereas 2 is in the domain of -1/(x+3) but not in the other expression’s domain. Therefore 2 is not in the domain of both expressions. DIVIDING RATIONAL EXPRESSIONS • To divide two rational expressions, we simply change the operation to multiplication using the reciprocal of the divisor. • Example: 𝑦3 −1 𝑦2 −1 𝑦2 +𝑦+1 𝑦2 +2𝑦+1 𝑦 3 −1 𝑦 2 +𝑦+1 = 2 ∙ 2 𝑦 −1 𝑦 +2𝑦+1 = y +1 ADDING OR SUBTRACTING • When rational expressions have the same denominator, we can add or subtract the numerators and retain the common denominator. • If the denominators differ, we must find equivalent rational expressions that have a common denominator. • In general, it is most efficient to find the least common denominator of the two expressions. EXAMPLE • 𝑥 2 −4𝑥+4 2𝑥 2 −3𝑥+1 + 𝑥+4 2𝑥−2 COMPLEX RATIONAL EXPRESSIONS • A complex rational expression has rational expressions in its numerator or its denominator or both. • To simplify a complex rational expression • Method 1: Find the LCD of all the denominators within the complex rational expression. Then multiply by 1 using the LCD as the numerator and the denominator of the expression for 1. • Method 2: First add or subtract, if necessary, to get a single rational expression in the numerator and in the denominator. Then divide by multiplying by the reciprocal of the denominator METHOD 1 • Simplify: 1 1 + 𝑎 𝑏 1 1 + 𝑎3 𝑏3 METHOD 2 • Simplify: 1 1 + 𝑎 𝑏 1 1 + 𝑎3 𝑏3 R.6 • Radical Notation and • Rational Exponents NTH ROOT • A number c is said to be an nth root if 𝑐 𝑛 = a • A number c is said to be a square root of a if c times c is equal to a. • Thus 3 is a square root of 9 because 3 times 3 is nine • -3 is also a square root of 9 • Similarly, 5, is a third root (called “cube root”) of 125 because 5 times 5 times 5 is 125. • This number has no other real-number cube roots as (-5) cubed would give -125. • Vocabulary: the symbol is called a radical and the expression is called a radicand. The number n is called the index. The positive root of an even radical is called the principal root. PROPERTIES OF RADICALS APPLICATION • Pythagorean Theorem • The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse: a² + b² = c² RATIONALIZING DENOMINATORS AND NUMERATORS • There are times when we need to remove the radicals in a denominator or a numerator. This is called rationalizing the denominator or rationalizing the numerator. • It is done by multiplying by one in such a way as to obtain a perfect nth power. • Rationalize the numerator: 𝑥− 𝑦 5 RATIONAL EXPONENTS • We are motivated to define rational exponents so that the properties for integer exponents hold for them as well. PROBLEM SETS R. 5 R. 6 #5-70 #4-100 5s 4s THE BASICS TO EQUATION SOLVING • An equation is a statement that two expressions are equal. To solve an equation in one variable is to find all the values of the variable that make the equation true. • Each of these numbers is called a solution of the equation. The set of all solutions of an equation is its solution set. • Equations that have the same solution set are called equivalent equations. LINEAR AND QUADRATIC EQUATIONS • A linear equation in one variable is an equation that is equivalent to one of the form ax + b = 0, where a and b are real numbers and a is not equal to zero. • A quadratic equation is an equation that is equivalent to one of the form ax² + bx + c = 0, where a, b, and c are real numbers and a is not equal to zero. EQUATION SOLVING PRINCIPLES • For any real numbers a, b, and c, • The Addition Principle: If a = b is true then a + c = b + c is also true. • The Multiplication Principle: If a = b is true then ac = bc is also true. • The Principle of Zero Products: If ab = 0 is true, then a = 0 or b = 0 and if a = 0 or b = 0 the ab = 0 • The Principle of Square Roots: If x² = k, then x = 𝑘 or x = - 𝑘 EXAMPLES • Solve: 2x + 3 = 1 – 6(x – 1) • Solve: x² - 3x = 4 PROBLEM SETS R. 7 #2-54 Evens Review Ch. R #2-78 Evens