Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2


 Corey Willis Jennings
 3 years ago
 Views:
Transcription
1 Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is a number, a variable, or a product of numbers and variables. A polynomial is a variable expression in which the terms are monomials. A polynomial of two terms is a binomial. A polynomial of three terms is a trinomial. The degree of a polynomial in one variable is the value of the largest exponent on the variable. The degree of 4x 3 3x 2 + 6x 1 is 3, the degree of 5y 4 2y 3 + y 2 7y + 8 is 4, the degree of a nonzero constant (number) is 0, and the number zero has no degree. Polynomials can be added using a vertical or horizontal format. The opposite of a polynomial is the polynomial with the sign of every term changed, i.e., (x 2 2x + 3) = x 2 + 2x 3. Chapter 10 Section 2: Multiplication of Monomials Rule for Multiplying Exponential Expressions If m and n are integers,then x m x n = x m+n. Rule for Simplifying Powers of Exponential Expressions If m and n are integers, then (x m ) n = x mn Rule for Simplifying Powers of Products If m, n, and p are integers, then (x m y n ) p = x mp y np 1
2 Chapter 10 Section 3: Multiplication of Polynomials FOIL Method (A + B) (C + D) = AC + AD + BC + BD Product of the Sum and Difference of Two Terms The Square of a Binomial (a + b) (a b) = a 2 b 2 (a + b) 2 = (a + b) (a + b) = a 2 + 2ab + b 2 (a b) 2 = (a b) (a b) = a 2 2ab + b 2 Chapter 10 Section 4 Integer Exponents and Scientific Notation Zero as an Exponent If x 0, then x 0 = 1. The expression 0 0 is not defined. Definition of Negative Exponents If n is a positive integer and x 0, then x n = 1 x n and 1 x n. Rules for Dividing Exponential Expressions If m and n are integers and x 0, then xm x n = xm n. Chapter 10 Section 5: Division of Polynomials We divide polynomials using a method similar to the method used for the division of whole numbers. 2
3 Figure 1: Dividing Whole Numbers Figure 2: Dividing Polynomials Chapter 11 Section 1: Common Factors The greatest common factor (GCF) of two or more monomials is the greatest integer and the variable with the smallest exponent that is a factor of each monomial. For example the GCF of x 2, x 4, and x 6 is x 2 because 2 is the smallest exponent of the three monomials. To factor a polynomial means to write the polynomial as a product of other polynomials. Chapter 11 Section 2: Form x 2 + bx + c Factoring Polynomials of the To factor polynomials of the form x 2 + bx + c we ask ourselves What two numbers multiply to give c AND add to give b? In other words, we seek integers m and n such that m n = c and m + n = b. Then the polynomial x 2 + bx + c can be factored as (x + m) (x + n). Thus, x 2 + bx + c = (x + m) (x + n). 3
4 Chapter 11 Section 3: Form ax 2 + bx + c Factoring Polynomials of the One method we can use to factor polynomials of the form ax 2 +bx+c is called the trial and error method. To use the trial and error method, we use the factors of a and the factors of c. Figure 3: Trial and Error Method Another (faster) method we can use to factor polynomials of the form ax 2 + bx + c is called the ac method or the factor by grouping method. To use factor by grouping, we multiply a c and look for two numbers m and n so that a c = m n and m + n = b. Then we will split the middle term into mx + nx so that the polynomial ax 2 + bx + c becomes ax 2 + mx + nx + c. Remember, we can only use factor by grouping on a polynomial that has four terms. Chapter 11 Section 4 Special Factoring Factoring the Difference of Two Squares a 2 b 2 = (a + b)(a b) Factoring a PerfectSquare Trinomial a 2 + 2ab + b 2 = (a + b) (a + b) = (a + b) 2 a 2 2ab + b 2 = (a b) (a b) = (a b) 2 4
5 Factoring the Sum or Difference of Two Perfect Cubes a 3 + b 3 = (a + b) ( a 2 ab + b 2) a 3 b 3 = (a b) ( a 2 + ab + b 2) Certain trinomials that are not quadratic can be expressed as a quadratic trinomials by making suitable variable substitutions. A trinomial is quadratic in form if it can be written as au 2 + bu + c. General Factoring Strategy When factoring a polynomial completely, ask the following questions about the polynomial. 1. Is there a common factor? If so, factor out the GCF. 2. If the polynomial is a binomial, is it the difference of two perfect squares, the sum of two perfect cubes, or the difference of two perfect cubes? If so, factor. 3. If the polynomial is a trinomial, is it a perfectsquare trinomial or the product of two binomials? If so, factor. 4. Can the polynomial be factored by grouping? If so, factor. 5. Is each factor nonfactorable over the integers? If not, factor. Chapter 11 Section 5: Solving Equations Principle of Zero Products If the product of two (or more) factors is zero, then at least one of the factors must be zero. If a b = 0 then a = 0 or b = 0. Steps in Solving a Quadratic Equation by Factoring 1. Write the equation in standard form. 2. Factor the polynomial. 3. Set each factor equal to zero. 4. Solve each equation for the variable. 5. Check the solutions. Chapter 12 Section 1: Rational Expressions Multiplication and Division of A rational expression is a fraction in which the numerator and the denominator are polynomials. A rational expression is in simplest form when the numerator and the denominator have no common factors other than 1. To simplify the rational expression we use 5
6 the property A C B C = A B. Multiply Rational Expressions To multiply rational expressions we use the property a b c d = ac bd. Divide Rational Expressions To divide rational expressions we use the property a b c d = a b d c Figure 4: Dividing Rational Expressions Using Keep Change Flip Chapter 12 Section 2: Addition and Subtraction of Rational Expressions To find the least common multiple (LCM) of two or more polynomials, first factor each polynomial completely. The LCM is the product of each factor the greatest number of times it occurs in any one factorization. Adding and Subtracting Rational Expressions with Common Denominators To add or subtract rational expressions with the same denominator, we use the properties a b + c b = a + c or a b b c b = a c b Adding and Subtraction Rational Expressions without Common Denominators To add or subtract rational expressions without the same denominator, we find the least common multiple of the denominators and express each fraction in terms of the common denominator. 6
7 Below are some examples for us to try with solutions at the end. 1. Simplify ( 3ab) 2 ( 2ab) Multiply ab (4a 2 3ab 8b 2 ). 3. Multiply (3x 7y) (3x + 5y). 4. Multiply (6x 4y) Simplify 22a4 b 8 c 4 33a 7 b 5 c 6. Write the number in scientific notation. 7. Write the number 7, 320, 000 in scientific notation. 8. Divide 24a2 b + 3ab 21ab 2. 3ab 9. Divide (2x x + 7) (x + 7). 10. Factor 4a a. 11. Factor 3p 3 16p 2 + 5p by grouping. 12. Factor 1 64b Factor 2x 4 13x Factor 7x Solve the equation z 2 + 5z 14 = Multiply y2 + y 20 y 2 + 2y 15 y2 + 5y 24 y 2 + 4y Divide 6a2 y + 3a 2 2x 3 + 4x Simplify 2x 4 4y 2 x + 1 6xy. 12ay + 6a 6x x. 2 7
8 Solutions Solution to 1: ( 3ab) 2 ( 2ab) 3 = ( 3) 2 a 2 b 2 ( 2) 3 a 3 b 3 = 9a 2 b 2 ( 8)a 3 b 3 = 72a 2+3 b 2+3 = 72a 5 b 5 Solution to 2: ab(4a 2 3ab 8b 2 ) = ab(4a 2 ) + ab( 3ab) + ab( 8b 2 ) = 4a 2+1 b 3a 1+1 b 1+1 8ab 1+2 = 4a 3 b 3a 2 b 2 8ab 3 Solution to 3: (3x 7y) (3x + 5y) = (3x)(3x) + (3x)(5y) + ( 7y)(3x) + ( 7y)(5y) = 9x xy 21xy 35y 2 = 9x 2 6xy 35y 2 Solution to 4: (6x 4y) 2 = (6x 4y) (6x 4y) = (6x)(6x) + (6x)( 4y) + ( 4y)(6x) + ( 4y)( 4y) = 36x 2 24xy 24xy + 16y 2 = 36x 2 48xy + 16y 2 Solution to 5: 22a 4 b 8 c 4 33a 7 b 5 c = 2 11b8 5 c a 7 4 = 2b3 c 3 3a 3 Solution to 6: To write in scientific notation, we will move the decimal point to the right until there is only one nonzero number to the left of the decimal point. Hence, we will move the decimal point to the right 6 places. Since we are moving the decimal point to the right, the exponent of 10 will be negative =
9 Solution to 7: To write 7, 320, 000 in scientific notation, first note that 7, 320, 000 = 7, 320, We will move the decimal point to the left until there is only one nonzero number to the left of the decimal point. Hence, we will move the decimal point to the left 6 places. Since we are moving the decimal point to the left, the exponent of 10 will be positive. 7, 320, 000 = Solution to 8: We will use the laws of exponents along with the property a + b c = a c + b c. 24a 2 b + 3ab 21ab 2 3ab = 24a2 b 3ab + 3ab 3ab 21ab2 3ab = 8a b 2 1 = 8a + 1 7b Solution to 9: x + 7 ) 2x x + 7 2x 2 14x Solution to 10: 2x + 1Hence, the solution is 2x + 1. x + 7 x 7 0 4a a 2 + 8a = 4a(a 2 + 3a + 2) = 4a(a + 1)(a + 2) To factor a 2 + 3a + 2, we ask ourselves What two numbers multiply to give 2 and add to give 1? Note that 2 1 = 2 AND = 3. Solution to 11: 3p 3 16p 2 + 5p = p(3p 2 16p + 5) = p(3p 2 15p + p + 5) = p [ (3p 2 15p) + ( p + 5) ] = p [3p(p 5) + 1(p 5)] = p (p 5) (3p 1) Solution to 12: Note: 1 3 = 1 and 64b 3 = (4b) 3. Hence, this polynomial is a difference of cubes. Thus, we can use the difference of cubes formula a 3 b 3 = (a b)(a 2 + ab + b 2 ) with a = 1 and b = 4b. 9
10 1 64b 3 = 1 3 (4b) 3 = (1 4b)( (4b) + (4b) 2 ) = (1 4b)(1 + 4b + 16b 2 ) = (1 4b)(16b 2 + 4b + 1) Solution to 13: This polynomial is quadratic in form. By letting u = x 2, u 2 = (x 2 ) 2 = x 4, the polynomial 2x 4 13x 2 15 becomes 2u 2 13u 15. Once we have factored this polynomial, we will replace u with x 2. 2x 4 13x 2 15 = 2u 2 13u 15 = 2u 2 15u + 2u 15 = (2u 2 15u) + (2u 15) = u(2u 15) + 1(2u 15) = (u + 1)(2u 15) = (x 2 + 1)(2x 2 15) Solution to 14: We will use the difference of squares formula a 2 b 2 = (a b)(a + b). 7x = 7(x 2 81) = 7(x ) = 7(x 9)(x + 9) Solution to 15: To solve this equation, we will factor the left hand side and then use the zero product property. To factor the left hand side, we ask ourselves What two numbers multiply to give 14 and add to give 5? Note that 7 ( 2) = 14 and 7 + ( 2) = 5. z 2 + 5z 14 = 0 (z + 7)(z 2) = 0 z + 7 = 0z 2 = 0 z = 7z = 2 Solution to 16: To multiply two rational expressions, we factor each numerator and denominator and use the property A C B C = A B. 10
11 y 2 + y 20 y 2 + 2y 15 y2 + 5y 24 y 2 + 4y 32 Solution to 17: (y + 5)(y 4) (y + 8)(y 3) = (y + 5)(y 3) (y + 8)(y 4) (y + 5)(y 4)(y + 8)(y 3) = (y + 5)(y 3)(y + 8)(y 4) = 1 6a 2 y + 3a 2 2x 3 + 4x 2 12ay + 6a 6x x = 6a2 y + 3a 2 2 2x 3 + 4x 6x3 + 12x ay + 6a = 3a2 (2y + 1) 2x 2 (x + 2) 6x2 (x + 2) 6a(2y + 1) = 3a2 6x 2 (2y + 1)(x + 2) 6a 2x 2 (x + 2)(2y + 1) = 3a2 6x 2 6a 2x 2 = 3a2 2a = 3a2 1 2 = 3a 2 Solution to 18: First, we must find the LCM of 4y 2 and 6xy. To find the LCM, we use each variable with the largest exponent and the largest product of each prime number. Hence, the LCM of 4y 2 and 6xy is x y 2 = 12xy 2 Note that 4y 2 3x = 12xy 2 and 6xy 2y = 12xy 2. 2x 4 x + 1 4y 2 6xy = 2x 4 4y 2 = 3x 3x x + 1 2y 6xy 2y 2y(x + 1) 12xy 2 3x(2x 4) 12xy 2 = 6x2 12x 12xy 2 2xy + 2y 12xy 2 = 6x2 12x 2xy 2y 12xy 2 = 3x2 6x xy y 6xy 2 11
Multiplication of Polynomials
Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is
More information5.1 Monomials. Algebra 2
. Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific
More information5.3. Polynomials and Polynomial Functions
5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a
More informationTopic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3
Topic 7: Polynomials Table of Contents 1. Introduction to Polynomials. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Special Products of Binomials 5. Factoring Polynomials 6. Factoring
More informationAlgebra I Polynomials
Slide 1 / 217 Slide 2 / 217 Algebra I Polynomials 20140424 www.njctl.org Slide 3 / 217 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying
More informationNever leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!
1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a
More informationSection September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc.
Section 2.12.2 September 6, 2017 1 Polynomials Definition. A polynomial is an expression of the form a n x n + a n 1 x n 1 + + a 1 x + a 0 where each a 0, a 1,, a n are real numbers, a n 0, and n is a
More informationAlgebra I Unit Report Summary
Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit  ( Ascend Default unit) 1. A01_01_01 HAB.1 Word Phrases As Algebraic Expressions 2. A01_01_02
More informationAdding and Subtracting Polynomials
Adding and Subtracting Polynomials Polynomial A monomial or sum of monomials. Binomials and Trinomial are also polynomials. Binomials are sum of two monomials Trinomials are sum of three monomials Degree
More information27 Wyner Math 2 Spring 2019
27 Wyner Math 2 Spring 2019 CHAPTER SIX: POLYNOMIALS Review January 25 Test February 8 Thorough understanding and fluency of the concepts and methods in this chapter is a cornerstone to success in the
More informationAlgebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials
Algebra 1: Hutschenreuter Chapter 10 Notes Name 10.1 Adding and Subtracting Polynomials Polynomial an expression where terms are being either added and/or subtracted together Ex: 6x 4 + 3x 3 + 5x 2 +
More informationChapter 5: Exponents and Polynomials
Chapter 5: Exponents and Polynomials 5.1 Multiplication with Exponents and Scientific Notation 5.2 Division with Exponents 5.3 Operations with Monomials 5.4 Addition and Subtraction of Polynomials 5.5
More informationAlgebra I. Book 2. Powered by...
Algebra I Book 2 Powered by... ALGEBRA I Units 47 by The Algebra I Development Team ALGEBRA I UNIT 4 POWERS AND POLYNOMIALS......... 1 4.0 Review................ 2 4.1 Properties of Exponents..........
More informationAlgebra I. Polynomials.
1 Algebra I Polynomials 2015 11 02 www.njctl.org 2 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying
More informationChapter Six. Polynomials. Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring
Chapter Six Polynomials Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring Properties of Exponents The properties below form the basis
More informationA2. Polynomials and Factoring. Section A2 1
A Polynomials and Factoring Section A 1 What you ll learn about Adding, Subtracting, and Multiplying Polynomials Special Products Factoring Polynomials Using Special Products Factoring Trinomials Factoring
More informationStudy Guide for Math 095
Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.
More informationReview for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.
LESSON 6 Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in
More informationMathB65 Ch 4 VII, VIII, IX.notebook. November 06, 2017
Chapter 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest
More informationUNIT 2 FACTORING. M2 Ch 11 all
UNIT 2 FACTORING M2 Ch 11 all 2.1 Polynomials Objective I will be able to put polynomials in standard form and identify their degree and type. I will be able to add and subtract polynomials. Vocabulary
More informationMath 75 MiniMod Due Dates Spring 2016
MiniMod 1 Whole Numbers Due: 4/3 1.1 Whole Numbers 1.2 Rounding 1.3 Adding Whole Numbers; Estimation 1.4 Subtracting Whole Numbers 1.5 Basic Problem Solving 1.6 Multiplying Whole Numbers 1.7 Dividing
More information1.3 Algebraic Expressions. Copyright Cengage Learning. All rights reserved.
1.3 Algebraic Expressions Copyright Cengage Learning. All rights reserved. Objectives Adding and Subtracting Polynomials Multiplying Algebraic Expressions Special Product Formulas Factoring Common Factors
More informationBeginning Algebra. 1. Review of PreAlgebra 1.1 Review of Integers 1.2 Review of Fractions
1. Review of PreAlgebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:
More informationUnit 21: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions
CP Algebra Unit 1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.
More information{ independent variable some property or restriction about independent variable } where the vertical line is read such that.
Page 1 of 5 Introduction to Review Materials One key to Algebra success is identifying the type of work necessary to answer a specific question. First you need to identify whether you are dealing with
More informationMULTIPLYING TRINOMIALS
Name: Date: 1 Math 2 Variable Manipulation Part 4 Polynomials B MULTIPLYING TRINOMIALS Multiplying trinomials is the same process as multiplying binomials except for there are more terms to multiply than
More informationMathB65 Ch 4 IV, V, VI.notebook. October 31, 2017
Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest
More informationx 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?
1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number
More informationRadicals: To simplify means that 1) no radicand has a perfect square factor and 2) there is no radical in the denominator (rationalize).
Summer Review Packet for Students Entering Prealculus Radicals: To simplify means that 1) no radicand has a perfect square factor and ) there is no radical in the denominator (rationalize). Recall the
More informationMath 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS:
Math 2 Variable Manipulation Part 2 Powers & Roots PROPERTIES OF EXPONENTS: 1 EXPONENT REVIEW PROBLEMS: 2 1. 2x + x x + x + 5 =? 2. (x 2 + x) (x + 2) =?. The expression 8x (7x 6 x 5 ) is equivalent to?.
More informationName: Chapter 7: Exponents and Polynomials
Name: Chapter 7: Exponents and Polynomials 71: Integer Exponents Objectives: Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents. You
More informationMath 0320 Final Exam Review
Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:
More informationWhen you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut.
Squaring a Binomial When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut. Solve. (x 3) 2 Step 1 Square the first term. Rules
More informationLesson 6. Diana Pell. Monday, March 17. Section 4.1: Solve Linear Inequalities Using Properties of Inequality
Lesson 6 Diana Pell Monday, March 17 Section 4.1: Solve Linear Inequalities Using Properties of Inequality Example 1. Solve each inequality. Graph the solution set and write it using interval notation.
More informationUnit 13: Polynomials and Exponents
Section 13.1: Polynomials Section 13.2: Operations on Polynomials Section 13.3: Properties of Exponents Section 13.4: Multiplication of Polynomials Section 13.5: Applications from Geometry Section 13.6:
More informationModule 3 Study Guide. GCF Method: Notice that a polynomial like 2x 2 8 xy+9 y 2 can't be factored by this method.
Module 3 Study Guide The second module covers the following sections of the textbook: 5.45.8 and 6.16.5. Most people would consider this the hardest module of the semester. Really, it boils down to your
More informationSYMBOL NAME DESCRIPTION EXAMPLES. called positive integers) negatives, and 0. represented as a b, where
EXERCISE A1 Things to remember: 1. THE SET OF REAL NUMBERS SYMBOL NAME DESCRIPTION EXAMPLES N Natural numbers Counting numbers (also 1, 2, 3,... called positive integers) Z Integers Natural numbers, their
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 Order of Operations Expression Variable Coefficient
More informationMath Lecture 18 Notes
Math 1010  Lecture 18 Notes Dylan Zwick Fall 2009 In our last lecture we talked about how we can add, subtract, and multiply polynomials, and we figured out that, basically, if you can add, subtract,
More informationSection 6.5 A General Factoring Strategy
Difference of Two Squares: a 2 b 2 = (a + b)(a b) NOTE: Sum of Two Squares, a 2 b 2, is not factorable Sum and Differences of Two Cubes: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 b 3 = (a b)(a 2 + ab + b
More informationSummer Prep Packet for students entering Algebra 2
Summer Prep Packet for students entering Algebra The following skills and concepts included in this packet are vital for your success in Algebra. The Mt. Hebron Math Department encourages all students
More informationAlgebra 2. Factoring Polynomials
Algebra 2 Factoring Polynomials Algebra 2 Bell Ringer MartinGay, Developmental Mathematics 2 Algebra 2 Bell Ringer Answer: A MartinGay, Developmental Mathematics 3 Daily Learning Target (DLT) Tuesday
More informationFactoring Polynomials. Review and extend factoring skills. LEARN ABOUT the Math. Mai claims that, for any natural number n, the function
Factoring Polynomials GOAL Review and extend factoring skills. LEARN ABOUT the Math Mai claims that, for any natural number n, the function f (n) 5 n 3 1 3n 2 1 2n 1 6 always generates values that are
More informationIntermediate Algebra Textbook for Skyline College
Intermediate Algebra Textbook for Skyline College Andrew Gloag Anne Gloag Mara Landers Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org To access a customizable
More informationA2T. Rational Expressions/Equations. Name: Teacher: Pd:
AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 13 HW: Pages 3 in Packet o Day : SWBAT: Factor using the Greatest
More informationPrerequisites. Copyright Cengage Learning. All rights reserved.
Prerequisites P Copyright Cengage Learning. All rights reserved. P.4 FACTORING POLYNOMIALS Copyright Cengage Learning. All rights reserved. What You Should Learn Remove common factors from polynomials.
More informationCan there be more than one correct factorization of a polynomial? There can be depending on the sign: 2x 3 + 4x 2 6x can factor to either
MTH95 Day 9 Sections 5.5 & 5.6 Section 5.5: Greatest Common Factor and Factoring by Grouping Review: The difference between factors and terms Identify and factor out the Greatest Common Factor (GCF) Factoring
More informationMA094 Part 2  Beginning Algebra Summary
MA094 Part  Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page
More informationPart 2  Beginning Algebra Summary
Part  Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian
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 informationLESSON 7.2 FACTORING POLYNOMIALS II
LESSON 7.2 FACTORING POLYNOMIALS II LESSON 7.2 FACTORING POLYNOMIALS II 305 OVERVIEW Here s what you ll learn in this lesson: Trinomials I a. Factoring trinomials of the form x 2 + bx + c; x 2 + bxy +
More informationQuadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents
Slide 1 / 200 Quadratic Functions Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations
More informationQuadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.
Slide 1 / 200 Quadratic Functions Slide 2 / 200 Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic
More informationSlide 1 / 200. Quadratic Functions
Slide 1 / 200 Quadratic Functions Key Terms Slide 2 / 200 Table of Contents Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic
More informationChapter 3: Section 3.1: Factors & Multiples of Whole Numbers
Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers Prime Factor: a prime number that is a factor of a number. The first 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
More informationMATH98 Intermediate Algebra Practice Test Form A
MATH98 Intermediate Algebra Practice Test Form A MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation. 1) (y  4)  (y + ) = 3y 1) A)
More informationMATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline
MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline 1. Real Numbers (33 topics) 1.3 Fractions (pg. 27: 175 odd) A. Simplify fractions. B. Change mixed numbers
More informationReview Notes  Solving Quadratic Equations
Review Notes  Solving Quadratic Equations What does solve mean? Methods for Solving Quadratic Equations: Solving by using Square Roots Solving by Factoring using the Zero Product Property Solving by Quadratic
More informationEvaluate the expression if x = 2 and y = 5 6x 2y Original problem Substitute the values given into the expression and multiply
Name EVALUATING ALGEBRAIC EXPRESSIONS Objective: To evaluate an algebraic expression Example Evaluate the expression if and y = 5 6x y Original problem 6() ( 5) Substitute the values given into the expression
More informationMidterm 3 Review. Terms. Formulas and Rules to Use. Math 1010, Fall 2011 Instructor: Marina Gresham. Odd Root ( n x where n is odd) Exponent
Math 1010, Fall 2011 Instructor: Marina Gresham Terms Midterm 3 Review Exponent Polynomial  Monomial  Binomial  Trinomial  Standard Form  Degree  Leading Coefficient  Constant Term Difference of
More informationSections 7.2, 7.3, 4.1
Sections 7., 7.3, 4.1 Section 7. Multiplying, Dividing and Simplifying Radicals This section will discuss the rules for multiplying, dividing and simplifying radicals. Product Rule for multiplying radicals
More informationI CAN classify polynomials by degree and by the number of terms.
131 Polynomials I CAN classify polynomials by degree and by the number of terms. 131 Polynomials Insert Lesson Title Here Vocabulary monomial polynomial binomial trinomial degree of a polynomial 131
More informationPolynomials and Factoring
7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of
More informationGaithersburg High School Summer 2018 Math Packet For Rising Algebra 2/Honors Algebra 2 Students
Gaithersburg High School Math Packet For Rising Algebra 2/Honors Algebra 2 Students 1 This packet is an optional review of the skills that will help you be successful in Algebra 2 in the fall. By completing
More informationAdding and Subtracting Polynomials
Adding and Subtracting Polynomials When you add polynomials, simply combine all like terms. When subtracting polynomials, do not forget to use parentheses when needed! Recall the distributive property:
More informationSpring Nikos Apostolakis
Spring 07 Nikos Apostolakis Review of fractions Rational expressions are fractions with numerator and denominator polynomials. We need to remember how we work with fractions (a.k.a. rational numbers) before
More informationAlgebra I. Slide 1 / 216. Slide 2 / 216. Slide 3 / 216. Polynomials
Slide 1 / 216 Slide 2 / 216 lgebra I Polynomials 20151102 www.njctl.org Table of ontents efinitions of Monomials, Polynomials and egrees dding and Subtracting Polynomials Multiplying a Polynomial by
More informationmn 3 17x 2 81y 4 z Algebra I Definitions of Monomials, Polynomials and Degrees 32,457 Slide 1 / 216 Slide 2 / 216 Slide 3 / 216 Slide 4 / 216
Slide 1 / 216 Slide 2 / 216 lgebra I Polynomials 20151102 www.njctl.org Slide 3 / 216 Table of ontents efinitions of Monomials, Polynomials and egrees dding and Subtracting Polynomials Multiplying a
More informationArithmetic, Algebra, Number Theory
Arithmetic, Algebra, Number Theory Peter Simon 21 April 2004 Types of Numbers Natural Numbers The counting numbers: 1, 2, 3,... Prime Number A natural number with exactly two factors: itself and 1. Examples:
More informationSummer Mathematics Packet Say Hello to Algebra 2. For Students Entering Algebra 2
Summer Math Packet Student Name: Say Hello to Algebra 2 For Students Entering Algebra 2 This summer math booklet was developed to provide students in middle school an opportunity to review grade level
More informationVariables and Expressions
Variables and Expressions A variable is a letter that represents a value that can change. A constant is a value that does not change. A numerical expression contains only constants and operations. An algebraic
More informationFactor each expression. Remember, always find the GCF first. Then if applicable use the xbox method and also look for difference of squares.
NOTES 11: RATIONAL EXPRESSIONS AND EQUATIONS Name: Date: Period: Mrs. Nguyen s Initial: LESSON 11.1 SIMPLIFYING RATIONAL EXPRESSIONS Lesson Preview Review Factoring Skills and Simplifying Fractions Factor
More informationHONORS GEOMETRY Summer Skills Set
HONORS GEOMETRY Summer Skills Set Algebra Concepts Adding and Subtracting Rational Numbers To add or subtract fractions with the same denominator, add or subtract the numerators and write the sum or difference
More informationReal Numbers. Real numbers are divided into two types, rational numbers and irrational numbers
Real Numbers Real numbers are divided into two types, rational numbers and irrational numbers I. Rational Numbers: Any number that can be expressed as the quotient of two integers. (fraction). Any number
More informationAssignment #1 MAT121 Summer 2015 NAME:
Assignment #1 MAT11 Summer 015 NAME: Directions: Do ALL of your work on THIS handout in the space provided! Circle your final answer! On problems that your teacher would show work on be sure that you also
More informationGeometry 21 Summer Work Packet Review and Study Guide
Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the
More information5.1, 5.2, 5.3 Properites of Exponents last revised 6/7/2014. c = Properites of Exponents. *Simplify each of the following:
48 5.1, 5.2, 5.3 Properites of Exponents last revised 6/7/2014 Properites of Exponents 1. x a x b = x a+b *Simplify each of the following: a. x 4 x 8 = b. x 5 x 7 x = 2. xa xb = xa b c. 5 6 5 11 = d. x14
More informationreview To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = 17
1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient
More informationP.1: Algebraic Expressions, Mathematical Models, and Real Numbers
Chapter P Prerequisites: Fundamental Concepts of Algebra Precalculus notes Date: P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Algebraic expression: a combination of variables and
More informationA field trips costs $800 for the charter bus plus $10 per student for x students. The cost per student is represented by: 10x x
LEARNING STRATEGIES: Activate Prior Knowledge, Shared Reading, Think/Pair/Share, Note Taking, Group Presentation, Interactive Word Wall A field trips costs $800 for the charter bus plus $10 per student
More informationRead the following definitions and match them with the appropriate example(s) using the lines provided.
Algebraic Expressions Prepared by: Sa diyya Hendrickson Name: Date: Read the following definitions and match them with the appropriate example(s) using the lines provided. 1. Variable: A letter that is
More informationREAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y?
REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} REAL WORLD SCENARIOS 1. If 4x + y = 110 where 10 < x < 0, what is the least possible value of y? WORK AND ANSWER SECTION. Evaluate
More informationMath 10C Polynomials Concept Sheets
Math 10C Polynomials Concept Sheets Concept 1: Polynomial Intro & Review A polynomial is a mathematical expression with one or more terms in which the exponents are whole numbers and the coefficients
More informationCONTENTS COLLEGE ALGEBRA: DR.YOU
1 CONTENTS CONTENTS Textbook UNIT 1 LECTURE 11 REVIEW A. p. LECTURE 1 RADICALS A.10 p.9 LECTURE 1 COMPLEX NUMBERS A.7 p.17 LECTURE 14 BASIC FACTORS A. p.4 LECTURE 15. SOLVING THE EQUATIONS A.6 p.
More informationBeginning Algebra MAT0024C. Professor Sikora. Professor M. J. Sikora ~ Valencia Community College
Beginning Algebra Professor Sikora MAT002C POLYNOMIALS 6.1 Positive Integer Exponents x n = x x x x x [n of these x factors] base exponent Numerical: Ex:  = where as Ex: () = Ex:  = and Ex: () = Rule:
More informationWe say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:
R.4 Polynomials in one variable A monomial: an algebraic expression of the form ax n, where a is a real number, x is a variable and n is a nonnegative integer. : x,, 7 A binomial is the sum (or difference)
More information( ) Chapter 6 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3.
Chapter 6 Exercise Set 6.1 1. A prime number is an integer greater than 1 that has exactly two factors, itself and 1. 3. To factor an expression means to write the expression as the product of factors.
More informationProperties of Real Numbers
PreAlgebra Properties of Real Numbers Identity Properties Addition: Multiplication: Commutative Properties Addition: Multiplication: Associative Properties Inverse Properties Distributive Properties Properties
More informationTABLE OF CONTENTS. Introduction to Finish Line Indiana Math 10. UNIT 1: Number Sense, Expressions, and Computation. Real Numbers
TABLE OF CONTENTS Introduction to Finish Line Indiana Math 10 UNIT 1: Number Sense, Expressions, and Computation LESSON 1 8.NS.1, 8.NS.2, A1.RNE.1, A1.RNE.2 LESSON 2 8.NS.3, 8.NS.4 LESSON 3 A1.RNE.3 LESSON
More informationPolynomials and Polynomial Equations
Polynomials and Polynomial Equations A Polynomial is any expression that has constants, variables and exponents, and can be combined using addition, subtraction, multiplication and division, but: no division
More informationMATH Spring 2010 Topics per Section
MATH 101  Spring 2010 Topics per Section Chapter 1 : These are the topics in ALEKS covered by each Section of the book. Section 1.1 : Section 1.2 : Ordering integers Plotting integers on a number line
More informationLESSON 9.1 ROOTS AND RADICALS
LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical
More informationJUST THE MATHS UNIT NUMBER 1.5. ALGEBRA 5 (Manipulation of algebraic expressions) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.5 ALGEBRA 5 (Manipulation of algebraic expressions) by A.J.Hobson 1.5.1 Simplification of expressions 1.5.2 Factorisation 1.5.3 Completing the square in a quadratic expression
More informationFastTrack  MA109. Exponents and Review of Polynomials
FastTrack  MA109 Exponents and Review of Polynomials Katherine Paullin, Ph.D. Lecturer, Department of Mathematics University of Kentucky katherine.paullin@uky.edu Monday, August 15, 2016 1 / 25 REEF Question
More informationInstructor: Richard Getso Course: Math 200.P10 TR 1:00 PM Spring 2016 (Getso)
1/8/016 Practice Test 1 (Chapter 11) Richard Getso Student: Richard Getso Date: 1/8/16 Instructor: Richard Getso Course: Math 00.P10 TR 1:00 PM Spring 016 (Getso) Assignment: Practice Test 1 (Chapter 11)
More informationRising 8th Grade Math. Algebra 1 Summer Review Packet
Rising 8th Grade Math Algebra 1 Summer Review Packet 1. Clear parentheses using the distributive property. 2. Combine like terms within each side of the equal sign. Solving MultiStep Equations 3. Add/subtract
More informationGet Ready. 6. Expand using the distributive property. a) 6m(2m 4) b) 8xy(2x y) c) 6a 2 ( 3a + 4ab) d) 2a(b 2 6ab + 7)
Get Ready BLM 5 1... Classify Polynomials 1. Classify each polynomial by the number of terms. 2y x 2 + 3x + 2 c) 6x 2 y + 2xy + 4 d) x 2 + y 2 e) 3x 2 + 2x + y 4 6. Expand using the distributive property.
More informationDear Future PreCalculus Students,
Dear Future PreCalculus Students, Congratulations on your academic achievements thus far. You have proven your academic worth in Algebra II (CC), but the challenges are not over yet! Not to worry; this
More informationLesson 3: Polynomials and Exponents, Part 1
Lesson 2: Introduction to Variables Assessment Lesson 3: Polynomials and Exponents, Part 1 When working with algebraic expressions, variables raised to a power play a major role. In this lesson, we look
More informationPreCalculus Summer Packet Instructions
PreCalculus Summer Packet Instructions Dear Student, You are receiving this summer packet as a review of previously covered math topics needed to be successful in the upcoming math class you will be taking
More information