STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS


 Allison Morgan
 2 years ago
 Views:
Transcription
1 STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an important first step to begin the study of precalculus Simplifying polynomials A (simple) term is the product of real numbers and variables (with nonnegative exponents) The numerical factor is the coefficient of the term The degree of the term is the sum of the exponents which have variable bases A polynomial is a single term or the indicated sum of two or more terms The operations involving polynomials are illustrated below EXAMPLES: (see Chapter, pages 0 ), Simplify: (x  y)  x( y  x) x  y x  xy x x  y  xy x Multiply: (x )(x  ) x  0x  Recall that the method is: multiply the first terms of each binomial add (mentally if possible) the products of the outside and inside terms multiply the second terms in each binomial A special case of multiplication is squaring a binomial Squaring should be practiced until the product can be written down with no intermediate steps (unless the coefficients are large) (x ) 9x 0x where 9x (x) ; 0x (x)(); () The trinomial 9x 0x is called a perfect square trinomial and can be recognized as such because the first and third terms are perfect squares and the middle term is double the product of x and Simplify: a b  ab 9a b ab ab  a b Factoring Polynomials Factoring means to write a polynomial as a product of expressions The goal is to factor completely so that each factor is prime, that is, each factor cannot be factored further Always look first for a common monomial factor Then try the other factoring possibilities listed below i) Common monomial factoring (ALWAYS CHECK FOR A COMMON FACTOR FIRST!) ii) Trinomial factoring (trial and error method) iii) Difference of two squares: a  b (a b)(a  b) iv) Difference of two cubes: a  b (a  b)(a ab b ) v) Sum of two cubes: a b (a b)(a  ab b ) (The last three formulas above can be checked by multiplication) page
2 EXAMPLES : (see Chapter, pages  ) Factor completely 0x y  xy  y y( 0x  x  ) common factor of y y (?x )(?x ) possibilities are x & 0x; x & 0x: x & x y(x?)(x?) possibilities are: & ;  & ; & ;  & y(x )(x  checked by multiplication a  (a  )(a a 9) the difference of two cubes, see formula iv) above a b  9ab ab(a  9b ) common factor of ab ab(a b)(a  b) using the difference of two squares formula The Factor of (): It is always possible to factor out a () from any polynomial, resulting in every sign of the polynomial being reversed If the polynomial is a binomial such as x  y then x  y ()(x y) ()(y  x) Thus, if two terms are being subtracted, factoring out () merely reverses the order of the terms of the binomial EXAMPLES: a) x y x x(xy ) OR (x)(xy  ) b) a  b ()(b  a) Factoring by Grouping: (See Chapter, page ) Grouping may work to factor a polynomial of or more terms The goal is to group terms to create a common factor, a difference of two squares or some other familiar factoring form EXAMPLES: 9 a  x  x  a  (x x ) a  (x ) difference of two squares [a (x )][a  (x )] (a x )(a  x  ) 0 x  0x  9x x (x  )  (x  ) common factor of (x  ) (x  )(x  ) A Least Common Multiple (LCM) is the quantity used: ) as a lowest common denominator (LCD) when adding fractions; ) to clear fractions occurring in equations and some inequalities; ) to simplify complex fractions To find the LCM first factor each polynomial Then for each distinct factor pick the largest exponent (See Chapter, page ) EXAMPLES: Find the LCM of the following: x  9 (x )(x  ) The LCM is: (x )(x  )(x ) x x  (x  )(x ) x x (x x ) (x ) page
3 Polynomial Fractions or Rational Expressions (See Chapter, pages  9) Factoring is a key skill when working with rational expressions EXAMPLES Reduce: x  0x x first factor numerator and denominator completely () (x  x  ) (x  )(x ) ( x)(  x) ( x)(  x) cancel any like factors () Note that (x  ) ()(  x)! (x ) x A fraction such as a   a can always be reduced by factoring out a (), namely, a   a ()(  a)  a  Equivalently, canceling leaves a factor of () in the numerator and of in the denominator Divide:  x x  x  x  9 x  Invert the divisor and multiply  x x  x  x  x  9 Factor top and bottom completely (  x) (x  ) (x )(x  ) (x )(x  ) () () (  x) (x  ) (x )(x  ) (x )(x  ) () () cancel any like factors  (x )(x ) Add (or subtract): a  a   a a   a Find the LCD  a (a  )  a ( a)(  a) Do not include both (a  ) & (  a) in the LCD (a  ) a ( a)(  a)  (  a)  a ( a)(  a) LCD: (  a)( a)  (  a) a a  a ( a)(  a) ( a) (  a) (  a)( a)   a  a (  a)( a)  a (  a)( a) page
4 COMPLEX FRACTIONS (See Chapter, pages 90) A complex fraction is a fraction that has fractions in the numerator or in the denominator or both Complex fractions will occur frequently throughout this course A recommended method of simplifying is to multiply by a carefully selected name for one, that is, simplify by multiplying top and bottom by the same expression EXAMPLES: Simplify:  The, & are the "extra" denominators of the complex fraction The LCM of, & is Multiply by to simplify this complex fraction With practice the second step above can be eliminated Simplify: x  x  (x ) (x )  (x ) (x )(x  )  x  (x )(x  )  x (x )(x  ) (  x) (x )(x  )  (x ) OR (x ) Simplifying Radicals: (See Chapter, pages  ) The radicand of 9 is 9 and the order of the radical is Simplifying radicals means to change to an equivalent form: i) with no perfect nth power factors in the radicand (order n radicals), and ii) with an integer radicand (or sometimes no radical in the denominator) EXAMPLES: Simplify a) 9 9 b) Remember to cancel like factors before multiplying! page
5 a) 0 0 OR 0 b) Operations with Radicals: (See Chapter, pages  ) Radicals with the same radicand and same order are like radicals Add or subtract coefficients of like radicals Radicals of the same order can be multiplied by multiplying the radicands Similarly, radicals of the same order can be divided by dividing the radicands EXAMPLES: Perform the indicated operations 9 First simplify each radical 0 ( ) (  )   (Similar to polynomial multiplication!)   If the denominator has two or more terms, the simplified form will be an equivalent form with a rational number denominator (actually the denominator will become an integer) Multiply top and bottom by the conjugate to simplify Note that (  ) ( ) has the form of multiplying (a b)(a  b) a  b Imaginary or Complex Numbers (See Chapter, pages  ) Expressions such as  do not represent real numbers since no real number squared equals  In this case,  9 () i where i , an imaginary number page
6 EXAMPLES: Simplify i  i (  i) (  i) OR  i Radicals and Exponents (See Chapter, pages  ) Recall that: b 0, b 0;  ; / ; / Also, for x 0, x a x b x ab ( xy ) x a x b x ab x y ( x a ) b x ab a a x a y a xa y a As noted previously it is not possible to multiply because the orders are different However, using exponential forms sometimes allows some simplifying EXAMPLES: Perform the indicated operations / / // / or or  (  9 / ) 9 / xy x  y  x y xy  x xy y Note that this was merely a complex fraction! y x x y Simplify and write the answer without negative or zero exponents x / y  / / x / y  / Simplify inside the parentheses first  /  (/) Subtract exponents of like bases: x // y (Note:   AND / 9  ) So, x / y  / x / y  / / x / y  / / / x / / y / / page
7 x / x / y / y / Linear Equations and Inequalities (See Chapter, pages , Chapter, page ) EXAMPLES: Solve x x  0 C lear fractions by multiplying each side by 0, the LCM of the denominators x 0 0 x  x 90x  Get all terms with the variable on one side and everything else on the other side 90x  x x So, x and the solution set is S 9 x Recall that statements of the form a x b are implied "AND" statements and mean that a x AND AT THE SAME TIME x b It is shortest to solve both at once as shown below x  x Clear fractions first (unless the denominators contain variables!) Add  to each part of the inequality to isolate the x term 9 x Divide each part by to solve for x 9 x 9 x IF YOU MULTIPLY OR DIVIDE AN INEQUALITY BY A NEGATIVE NUMBER, REMEMBER THAT THE INEQUALITY REVERSES! 0 Solve for c: abc bc  ab Isolate all terms that contain "c" ab bc  abc Factor out "c' from the right side ab c(b  ab) Divide each side by the coefficient of "c" ab ab a c b  ab b(  a)  a Completing the Square (See Chapter, page ) In addition to deriving the quadratic formula from ax bx c 0, a 0, completing the square has other important uses which will be encountered this semester as well as in future courses x   x : To complete the square, add  9 page
8 So, x  x 9 x  Quadratic Equations (See Chapter, pages  0) Quadratic equations have the form ax bx c 0, a 0 EXAMPLES: Solve (x )(x  ) x  9 Remove parentheses; x  x  0 x  9 x  x  0 Get a zero on one side Factor the nonzero side (x )(x  ) 0 Set each factor containing a variable equal to zero x 0 OR x  0 x OR x The solution set S, If the nonzero side cannot be factored using integers, then use the quadratic formula x b ± b  ac a and simplify x  x  0 The left side does not factor Use the quadratic formula with a, b , c  x () ± ()  ()() () ± ( ± ) ± ± ± Equations of the form x k for any real number k A special case of quadratic equations occurs when b 0, that is, the quadratic equation has the form x k for any real number k The most efficient method for solving these equations is to use the following x k then x ± k Notice that the square root is only on the right side!! In fact, x x for all real numbers x! x Special case of a quadratic equation x Solve for x x ± ± ± ± OR ± page
9 Equations with rational or radical expressions (See Chapter,  ) The goal is to convert to an equivalent equation that is either a familiar linear or quadratic form EXAMPLES: Solve x x(x ) x The LCM of the denominators is x(x ), x 0,  x(x ) x x(x ) x Clear fractions x x A linear equation; solve for x x  However, x  makes the LCM and a denominator equal to zero So, x  cannot be a solution and x  is called an extraneous root The solution set is the empty set, namely, S x  x Isolate the radical; if there are two radicals, isolate one of them x   x Square each side This could result in extra solutions! ( x  ) (  x ) x x x A quadratic equation 0 x  x (x  )(x  ) x OR x are POSSIBILITIES ONLY Each must be checked using the original equation Check x : Check x : Does ? Does ? 9 x is NOT a solution x is a solution The solution set is S {} page 9
10 Systems of Two Equations (See Chapter, pages  ) Systems of two or more equations can be solved by the Addition (Elimination) Method or by Substitution Each method is illustrated below EXAMPLES: Solve Use the Addition or Elimination Method x y x  y 0 Multiply the first equation by and the second equation by x y 9x  y 0 Add the equations to eliminate the y terms x x Substitute into either equation to find y Using the first equation () y y  y  The solution is the ordered pair (, ) The above system could be solved by substitution as well Use the Substitution Method y  x y x Solve the first equation for y and substitute into the second equation y x (x ) x, a quadratic equation in one variable x x 0 x  x  (x )(x  ) So, x  OR x Find the corresponding y value for each x value x  y  So, one solution is the ordered pair (, ) x y So, another solution is the ordered pair (, ) The solution set S {(, ), (, )} page 0
1.3 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 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 information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More informationMth 95 Module 2 Spring 2014
Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression
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 informationTool 1. Greatest Common Factor (GCF)
Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
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 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 informationChapter 4. Polynomials
4.1. Add and Subtract Polynomials KYOTE Standards: CR 8; CA 2 Chapter 4. Polynomials Polynomials in one variable are algebraic expressions such as 3x 2 7x 4. In this example, the polynomial consists of
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
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 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 informationAlgebra Tiles Activity 1: Adding Integers
Algebra Tiles Activity 1: Adding Integers NY Standards: 7/8.PS.6,7; 7/8.CN.1; 7/8.R.1; 7.N.13 We are going to use positive (yellow) and negative (red) tiles to discover the rules for adding and subtracting
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
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 information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationMyMathLab ecourse for Developmental Mathematics
MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationFactoring Polynomials
Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationMATH 90 CHAPTER 6 Name:.
MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
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 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 informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
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 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 informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationPOLYNOMIALS and FACTORING
POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
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 informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
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 informationPolynomials. 44 to 48
Polynomials 44 to 48 Learning Objectives 44 Polynomials Monomials, binomials, and trinomials Degree of a polynomials Evaluating polynomials functions Polynomials Polynomials are sums of these "variables
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
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 informationPre Cal 2 1 Lesson with notes 1st.notebook. January 22, Operations with Complex Numbers
0 2 Operations with Complex Numbers Objectives: To perform operations with pure imaginary numbers and complex numbers To use complex conjugates to write quotients of complex numbers in standard form Complex
More informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More informationA Systematic Approach to Factoring
A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool
More informationDevelopmental Math Course Outcomes and Objectives
Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/PreAlgebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and
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 informationChapter 4  Decimals
Chapter 4  Decimals $34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value  1.23456789
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationSection 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationAIP Factoring Practice/Help
The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There
More informationDeterminants can be used to solve a linear system of equations using Cramer s Rule.
2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution
More informationBasic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.
Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:
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 informationIn algebra, factor by rewriting a polynomial as a product of lowerdegree polynomials
Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting
More informationChapter R.4 Factoring Polynomials
Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x
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 informationCAHSEE on Target UC Davis, School and University Partnerships
UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,
More informationModule: Graphing Linear Equations_(10.1 10.5)
Module: Graphing Linear Equations_(10.1 10.5) Graph Linear Equations; Find the equation of a line. Plot ordered pairs on How is the Graph paper Definition of: The ability to the Rectangular Rectangular
More informationRadicals  Rationalize Denominators
8. Radicals  Rationalize Denominators Objective: Rationalize the denominators of radical expressions. It is considered bad practice to have a radical in the denominator of a fraction. When this happens
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationHFCC Math Lab Intermediate Algebra  7 FINDING THE LOWEST COMMON DENOMINATOR (LCD)
HFCC Math Lab Intermediate Algebra  7 FINDING THE LOWEST COMMON DENOMINATOR (LCD) Adding or subtracting two rational expressions require the rational expressions to have the same denominator. Example
More informationUnit: Polynomials and Factoring
Name Unit: Polynomials: Multiplying and Factoring Specific Outcome 10I.A.1 Demonstrate an understanding of factors of whole numbers by determining: Prime factors Greatest common factor Least common multiple
More information5.7 Literal Equations
5.7 Literal Equations Now that we have learned to solve a variety of different equations (linear equations in chapter 2, polynomial equations in chapter 4, and rational equations in the last section) we
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 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 informationMath 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:
Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?
More informationName Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
More informationUnit 3 Polynomials Study Guide
Unit Polynomials Study Guide 75 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial
More information27 = 3 Example: 1 = 1
Radicals: Definition: A number r is a square root of another number a if r = a. is a square root of 9 since = 9 is also a square root of 9, since ) = 9 Notice that each positive number a has two square
More informationA. Factoring out the Greatest Common Factor.
DETAILED SOLUTIONS AND CONCEPTS  FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
More informationRadicals  Multiply and Divide Radicals
8. Radicals  Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals
More information6.1 The Greatest Common Factor; Factoring by Grouping
386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.
More informationThis assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the
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 informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More information2.3. 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 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 informationThis is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0).
This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationName Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE
Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers
More informationEAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors.
EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an
More information2. Simplify. College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses
College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2
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 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 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 informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationLESSON 6.2 POLYNOMIAL OPERATIONS I
LESSON 6.2 POLYNOMIAL OPERATIONS I Overview In business, people use algebra everyday to find unknown quantities. For example, a manufacturer may use algebra to determine a product s selling price in order
More information3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes
Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationDifference of Squares and Perfect Square Trinomials
4.4 Difference of Squares and Perfect Square Trinomials 4.4 OBJECTIVES 1. Factor a binomial that is the difference of two squares 2. Factor a perfect square trinomial In Section 3.5, we introduced some
More informationFlorida Math Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies  Lower and Upper
Florida Math 0022 Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies  Lower and Upper Whole Numbers MDECL1: Perform operations on whole numbers (with applications,
More informationMATH 108 REVIEW TOPIC 10 Quadratic Equations. B. Solving Quadratics by Completing the Square
Math 108 T10Review Topic 10 Page 1 MATH 108 REVIEW TOPIC 10 Quadratic Equations I. Finding Roots of a Quadratic Equation A. Factoring B. Quadratic Formula C. Taking Roots II. III. Guidelines for Finding
More information6.4 Special Factoring Rules
6.4 Special Factoring Rules OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor a sum of cubes. By reversing the rules for multiplication
More informationFACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1
5.7 Factoring ax 2 bx c (549) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
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 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 informationALGEBRA I / ALGEBRA I SUPPORT
Suggested Sequence: CONCEPT MAP ALGEBRA I / ALGEBRA I SUPPORT August 2011 1. Foundations for Algebra 2. Solving Equations 3. Solving Inequalities 4. An Introduction to Functions 5. Linear Functions 6.
More informationA.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents
Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify
More information