Basic Properties of Rational Expressions


 Marianna Small
 5 years ago
 Views:
Transcription
1 Basic Properties of Rational Expressions A fraction is not defined when the denominator is zero! Examples: Simplify and use Mathematics Writing Style. a) x + 8 b) x 9 x 3 Solution: a) x + 8 (x + 4) x + 4 b) x 9 x 3 (x + 3)(x 3) x 3 x + 3
2 The Domain of a Rational Function: Examples: Find the domain of the following functions: a) f(x) x + 4 b) g(x) x + 5 x 4 c) h(x) x + 5 x 3 4x Solution: d) F(x) x x + 9 a) f(x) x + 4 This is a polynomial, and is defined for all values of x. So the domain is the set of real numbers. D R b) g(x) x + 5 x 4 This is a rational function and is not defined when the denominator is zero, or when x 4 0 x 4 So the domain is consists of all real numbers except that. c) h(x) x + 5 x 3 4x d) F(x) This is a rational function and is not defined when the denominator is zero, or when D {x R x 4 } x x + 9 This is a rational function and is not defined when the denominator is zero, or when x 3 4x 0 x x(x 4) 0 x(x )(x + ) 0 But this expression is never x 0,, So the domain is consists of all real numbers except these. zero for any real value of x. So the domain consists of all real numbers. D {x R x 0,, } D R
3 Multiplication and Division of Rational Expressions Example: Perform the indicated operation and simplify: x + x 3 x 3x 0 4x + x x x 9x 5 x x + x + x 3 x 3x 0 x x 4x + x 9x 5 x x + (x + 3)(x ) (x 5)(x + ) x(x ) (x + ) x(x + 3) (x + ) (x + )(x 5) (x )
4 Adding or Subtracting Fractions with Equal Denominators Examples: Perform the indicated operation and simplify: a) x x 4 + x 4 b) x x 4 4x x x 4 x + x 4 x (4x x ) x 4 x + (x + )(x ) x 4x + x x 4 x x 4x x 4 x(x ) (x + )(x ) x (x + )
5 Finding the LCD Step. Factor each denominator completely, including the prime factors of any constant factor. Step. Form the product of all the factors that appears in the complete factorizations. Step 3. The number of times any factors appears in the LCD is the most number of times it appears in any one factorization. Examples: Find the LCD for the given denominators: a) Denominators are 4, 30, and LCD b) Denominators are x 3 x and x 3 x x 3 x x (x ) x 3 x x(x ) x(x + )(x ) LCD x (x + )(x )
6 Adding or Subtracting Fractions with Unequal Denominators (FLEAS). Factor the rational expression.. Find the Least Common Denominator (LCD). 3. Equalize each denominators by replacing each fraction with an equivalent one whose denominator is the LCD. 4. Add or Subtract using RASFED. Example:
7 Examples: Perform the indicated operation and simplify. a) x x + x + b) x + x x + x x 4 (x + )(x ) (x + ) x + x x + x (x + )(x ) (x + ) (x + ) (x ) x (x + ) (x ) (x + ) (x + )(x ) x + x (x + )(x ) (x + ) (x ) (x + ) (x ) (x + 4x + 4) (x + x) (x + )(x ) x + x + (x + ) (x ) x + 4x + 4 x x (x + )(x ) x + 3 (x + ) (x ) x + 4 (x + )(x ) (x + ) (x + )(x ) (x )
8 Complex Fractions To simplify complex fractions: Step : Identify all fractions in the numerator and denominator and find the LCD. Step : Multiply the numerator and denominator by the LCD. Examples: 3 a) b) 7 xy x + y x + y x + y x + y xy (x + y) xy x + y xy (x + y) y + x
9 Example: Simplify the following: a) 9 y 3 y b) x + h x h y y 9 y 3 y 9y 3y y (3y + )(3y ) y(3y ) 3y + y x(x + h) x + h x x(x + h) h x (x + h) x(x + h) h x x h x(x + h) h h x(x + h) h x(x + h)
10 Long Division of Polynomials Monomial Denominator: When you divide a polynomial by a monomial, you must divide each term in the numerator by the denominator Examples: Perform the indicated operation. a) (x 3 6x + x) 3x b) Divide 5y 3 +0y 5y by 5y x3 6x + x 3x 5y 3 + 0y 5y 5y x3 3x 6x 3x + x 3x 5y 3 5y + 0y 5y 5y 5y x 3 x + 3 3y + 4y c) (6x 3 8x + 3x) x d) ( 6z 4 +6z 3 +8z +64z) 8z 6x3 8x + 3x x 6z4 +6z 3 +8z +64z 8z 6x3 x 8x 3x + x x 6z4 8z +6z3 8z + 8z 8z 64z 8z 8x 8x 3 + z 3 + z + z 8
11 Long Division of Polynomials Examples: Calculate the indicated quotients by long division: a) x3 x 7x + 3 x x 3 x 7x + 3 x + x 4x + + x + b) x4 8x 8 x x x 4 8x 8 x x + x + x 9 + x + 0 x x +
12 c) 6x4 + x 3 9x + 4 x + + 6x 4 + x 3 9x + 4 x 3x 3 + x + x Synthetic Division of Polynomials You can only use synthetic division when you divide a polynomial by a linear polynomial with linear coefficient. Examples: Calculate the indicated quotients by synthetic division: a) x3 x 7x + 3 x x 3 x 7x + 3 x + x 4x + + x +
13 b) x4 8x 8 x x 4 8x 8 x 3 x 3 + 3x + x x 3 c) x4 8 x x 4 8 x + 3 x 3 3x + 9x 7
14 Remainder Theorem When you divide a polynomial P(x) by the factor x c, the remainder is P(c). Thus we sometimes evaluate a polynomial P(x) when x c by performing the appropriate synthetic division. Examples : Let P(x) x 3 4x + 5. a) By direct substitution, evaluate P(). P(x) x 3 4x + 5 P() () 3 4() b) Find the remainder when P(x) is divided by x Remainder is 5 Examples : Let P(x) 4x 6 5x x 4 + 7x. Find P(4) P(4) 6 Note the problem is easier when we use the Remainder Theorem
15 Equations Involving Fractions To solve equations with (simple) fractions: Step : Identify all fractions in the equation and find the LCD. Step : Multiply the both sides of the equation by the LCD. Step 3: Solve the resulting equation. Step 4: Check the answer into the original problem. Examples: Solve the following: a) x 6 3 x 8 (x 6)(x 8) x 6 (x 6)(x 8) 3 x 8 (x 8) 3(x 6) x 6 3x 8 x x
16 b) z 4 z z z z 4 z 3 z z 4 z(z ) z z 4 z (z ) z (z ) z 4 z(z ) z (z ) z z 4 z (z ) z(z 4) z(z ) (z ) 4 z 4z z z z z 4z z 4z 0 0 The answer would be all real numbers, but we much check! all real numbers except 0, c) y y 3 y 9 y y 3 (y + 3)(y 3) (y + 3)(y 3) y y 3 (y + 3)(y 3) (y + 3)(y 3) (y + 3)(y ) (y + 3)(y 3) y + y 6 y 9 y 5
17 d) x 4 x + x x (x + )(x ) x + x(x ) x (x + )(x ) (x + )(x ) x (x + )(x ) x + x(x ) x (x + )(x ) + x(x + ) x x 4 + x + x x x + x 4 x x The only possible solution is x, but we must check! no solution
18 e) + 0 x + 3 x + 3 (x + )(x + 3) 0 + x + (x + )(x + 3) 3 x + 3 (x + )(x + 3) + 0(x + 3) 3(x + ) (x + 5x + 6) + 0x x + 6 x + 0x + + 0x x + 6 x + 7x (x + 9)(x + 4) 0 x 9/, 4 f) y + y y + 5 y + 6y + 3 y + y 6 (y + 3)(y ) y + y y + 5 y (y + 3)(y ) 6y (y+3)(y ) (y )(y + ) + (y + 3)(y + 5) (y + 3)(y ) + (6y + 3) y y + y + 8y + 5 y + y 6 + 6y + 3 y + 7y + 3 y + 7y + 7 y 4 y ± The solution would be y or y, but we must check! y
19 Example: It takes Rosa, traveling at 50 mph, 45 minutes longer to go a certain distance than it takes Maria traveling at 60 mph. Find the distance traveled. distance rate time Rosa x 50 x/50 Maria x 60 x/60 Important: 45 minutes 3/4 hours. We must use hours here! x 50 x x 50 x x 5x 5 x 5 5 miles
20 Example: Beth can travel 08 miles in the same length of time it takes Anna to travel 9 miles. If Beth s speed is 4 mph greater than Anna s, find both rates. Solution: distance rate time Beth 08 x /(x+4) Anna 9 x 9/x 08 x + 4 x(x + 4) 08 x x x(x + 4) 08 x x 9(x + 4) 08x 9x x 768 x 48 Beth 5 mph Anna 48 mph
21 Example: Toni needs 4 hours to complete the yard work. Her husband, Sonny, needs 6 hours to do the work. How long will the job take if they work together? Toni Sonny together 4 hours 6 hours x hours x x x x 3x + x 5x x 5 5 hours hours + 5 hours hours minutes hours 4 minutes
22 Example: Working together, Rick and Rod can clean the snow from the driveway in 0 minutes. It would have taken Rick, working alone, 36 minutes. How long would it have taken Rod alone? Rick 36 minutes Rod x minutes together 0 minutes 36 + x 0 80x 36 + x 80x 0 5x x 80 4x 45 x 45 minutes
23 Example: John, Ralph, and Denny, working together, can clean a store in 6 hours. Working alone, Ralph takes twice as long to clean the store as does John. Denny needs three times as long as does John. How long would it take each man working alone? John Ralph Denny together x hours x hours 3x hours 6 hours x + x + 3x 6 6x x + x + 3x 6x x x John minutes Ralph minutes Denny 33 minutes
24 Example: An inlet pipe on a swimming pool can be used to fill the pool in hours. The drain pipe can be used to empty the pool in 0 hours. If the pool is empty and the drain pipe is accidentally opened, how long will it take to fill the pool? inlet pipe drain pipe together hours 0 hours x hours 0 We subtract because the pipes are working against each other! 60x 0 x 60x x 5x 3x 60 x 60 x hours
25 Example: You can row, row, row your boat on a lake 5 miles per hour. On a river, it takes you the same time to row 5 miles downstream as it does to row 3 miles upstream. What is the speed of the river current in miles per hour? distance rate time downstream x 5/(5 + x) upstream 3 5 x 3/(5 x) x 3 5 x (5 + x)(5 x) x (5 + x)(5 x) 3 5 x 5(5 x) 3(5 + x) 5 5x 5 + 3x 0 8x 5 4 x 5 4 mph
2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationMarch 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More information317 1525 5 1510 25 32 5 0. 1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true
Section 5.2 solutions #110: a) Perform the division using synthetic division. b) if the remainder is 0 use the result to completely factor the dividend (this is the numerator or the polynomial to the
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationGraphing Rational Functions
Graphing Rational Functions A rational function is defined here as a function that is equal to a ratio of two polynomials p(x)/q(x) such that the degree of q(x) is at least 1. Examples: is a rational function
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationProcedure for Graphing Polynomial Functions
Procedure for Graphing Polynomial Functions P(x) = a n x n + a n1 x n1 + + a 1 x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine
More informationSimplifying Algebraic Fractions
5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions
More informationis identically equal to x 2 +3x +2
Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3
More information2.5 ZEROS OF POLYNOMIAL FUNCTIONS. Copyright Cengage Learning. All rights reserved.
2.5 ZEROS OF POLYNOMIAL FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions.
More informationRational Expressions  Least Common Denominators
7.3 Rational Expressions  Least Common Denominators Objective: Idenfity the least common denominator and build up denominators to match this common denominator. As with fractions, the least common denominator
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationPartial Fractions. p(x) q(x)
Partial Fractions Introduction to Partial Fractions Given a rational function of the form p(x) q(x) where the degree of p(x) is less than the degree of q(x), the method of partial fractions seeks to break
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationCollege Algebra  MAT 161 Page: 1 Copyright 2009 Killoran
College Algebra  MAT 6 Page: Copyright 2009 Killoran Zeros and Roots of Polynomial Functions Finding a Root (zero or xintercept) of a polynomial is identical to the process of factoring a polynomial.
More informationNegative Integer Exponents
7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions  that is, algebraic fractions  and equations which contain them. The reader is encouraged to
More information63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15.
9.4 (927) 517 Gear ratio d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases? decreases 100 80 60 40 20 27in. wheel, 44 teeth
More informationGreatest Common Factor (GCF) Factoring
Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationLimits and Continuity
Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationPolynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
More informationAlgebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials
More informationLagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.
Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method
More informationChapter 5. Rational Expressions
5.. Simplify Rational Expressions KYOTE Standards: CR ; CA 7 Chapter 5. Rational Expressions Definition. A rational expression is the quotient P Q of two polynomials P and Q in one or more variables, where
More informationPreCalculus II Factoring and Operations on Polynomials
Factoring... 1 Polynomials...1 Addition of Polynomials... 1 Subtraction of Polynomials...1 Multiplication of Polynomials... Multiplying a monomial by a monomial... Multiplying a monomial by a polynomial...
More information1.2 Linear Equations and Rational Equations
Linear Equations and Rational Equations Section Notes Page In this section, you will learn how to solve various linear and rational equations A linear equation will have an variable raised to a power of
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationFACTORING POLYNOMIALS
296 (540) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More information2.3 Solving Equations Containing Fractions and Decimals
2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions
More information3.2 The Factor Theorem and The Remainder Theorem
3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial
More informationChapter 7  Roots, Radicals, and Complex Numbers
Math 233  Spring 2009 Chapter 7  Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More information8 Polynomials Worksheet
8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions  Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationMATH 10550, EXAM 2 SOLUTIONS. x 2 + 2xy y 2 + x = 2
MATH 10550, EXAM SOLUTIONS (1) Find an equation for the tangent line to at the point (1, ). + y y + = Solution: The equation of a line requires a point and a slope. The problem gives us the point so we
More informationGouvernement du Québec Ministère de l Éducation, 2004 0400813 ISBN 2550435451
Gouvernement du Québec Ministère de l Éducation, 004 0400813 ISBN 550435451 Legal deposit Bibliothèque nationale du Québec, 004 1. INTRODUCTION This Definition of the Domain for Summative Evaluation
More information2.5 Zeros of a Polynomial Functions
.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the xaxis and
More informationSolving Rational Equations and Inequalities
85 Solving Rational Equations and Inequalities TEKS 2A.10.D Rational functions: determine the solutions of rational equations using graphs, tables, and algebraic methods. Objective Solve rational equations
More information1.1 Practice Worksheet
Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)
More informationExamples of Tasks from CCSS Edition Course 3, Unit 5
Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can
More informationSolving Linear Equations  Fractions
1.4 Solving Linear Equations  Fractions Objective: Solve linear equations with rational coefficients by multiplying by the least common denominator to clear the fractions. Often when solving linear equations
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
More informationAlgebra II Unit Number 4
Title Polynomial Functions, Expressions, and Equations Big Ideas/Enduring Understandings Applying the processes of solving equations and simplifying expressions to problems with variables of varying degrees.
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationis the degree of the polynomial and is the leading coefficient.
Property: T. HrubikVulanovic email: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 HigherDegree Polynomial Functions... 1 Section 6.1 HigherDegree Polynomial Functions...
More informationDefinitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).
Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32
More informationMTH 086 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 20, 2006
MTH 06 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 0, 006 Math 06, Introductory Algebra, covers the mathematical content listed below. In order
More informationSIMPLIFYING ALGEBRAIC FRACTIONS
Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationFree PreAlgebra Lesson 55! page 1
Free PreAlgebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationQuestion 2: How do you solve a matrix equation using the matrix inverse?
Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients
More informationMATH 90 CHAPTER 1 Name:.
MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.
More informationVeterans Upward Bound Algebra I Concepts  Honors
Veterans Upward Bound Algebra I Concepts  Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER
More informationALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section
ALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 53.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 64.2 Solving Equations by
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More informationOperations with Algebraic Expressions: Multiplication of Polynomials
Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the
More informationH/wk 13, Solutions to selected problems
H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More informationSummer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students
Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students Studies show that most students lose about two months of math abilities over the summer when they do not engage in
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationPartial Fractions Decomposition
Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationDifferentiation and Integration
This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationSOLVING POLYNOMIAL EQUATIONS
C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More information#6 Opener Solutions. Move one more spot to your right. Introduce yourself if needed.
1. Sit anywhere in the concentric circles. Do not move the desks. 2. Take out chapter 6, HW/notes #1#5, a pencil, a red pen, and your calculator. 3. Work on opener #6 with the person sitting across from
More informationa. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F
FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationACCUPLACER. Testing & Study Guide. Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011
ACCUPLACER Testing & Study Guide Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011 Thank you to Johnston Community College staff for giving permission to revise
More information(a) Write each of p and q as a polynomial in x with coefficients in Z[y, z]. deg(p) = 7 deg(q) = 9
Homework #01, due 1/20/10 = 9.1.2, 9.1.4, 9.1.6, 9.1.8, 9.2.3 Additional problems for study: 9.1.1, 9.1.3, 9.1.5, 9.1.13, 9.2.1, 9.2.2, 9.2.4, 9.2.5, 9.2.6, 9.3.2, 9.3.3 9.1.1 (This problem was not assigned
More informationFlorida Math 0018. Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies  Lower
Florida Math 0018 Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies  Lower Whole Numbers MDECL1: Perform operations on whole numbers (with applications, including
More information