Multiplication of Polynomials


 Erica Richardson
 3 years ago
 Views:
Transcription
1 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 the degree of the term. The degree of a polynomial is the largest degree of its terms. The term of a polynomial with the largest degree is the leading term. Its coefficient is the leading coefficient. A oneterm polynomial is a monomial. A twoterm polynomial is a binomial. A threeterm polynomial is a trinomial. Addition and Subtraction of Polynomials and Polynomial Functions 7y 4 2y 2 3y 8 is a polynomial with leading coefficient 7 and degree 4. f1x2 4x 3 6x 11 f is a polynomial function with leading term 4x 3 and leading coefficient 4. The degree of f is 3. For f 1x2 4x 3 6x 11, find f f To add or subtract polynomials, add or subtract like terms. Example 4 1 4x 3 y 3x 2 y x 3 y 5x 2 y 2 2 4x 3 y 3x 2 y 2 7x 3 y 5x 2 y 2 11x 3 y 8x 2 y 2 Section 5.2 Multiplication of Polynomials To multiply polynomials, multiply each term in the first polynomial by each term in the second polynomial. Special Products 1. Multiplication of conjugates 1x y21x y2 x 2 y 2 The product is called a difference of squares. 2. Square of a binomial 1x y2 2 x 2 2xy y 2 1x y2 2 x 2 2xy y 2 The product is called a perfect square trinomial. 1x 2213x 2 4x 112 3x 3 4x 2 11x 6x 2 8x 22 3x 3 10x 2 19x 22 13x 5213x 52 13x x y y y y 2 24y 9
2 392 Chapter 5 Polynomials Section 5.3 Division of Polynomials Division of polynomials: 1. For division by a monomial, use the properties a b c for c 0. a c b c and a b c a c b c 12a 2 6a 9 12a2 6a 9 4a 2 3 a 2. If the divisor has more than one term, use long division. 3. Synthetic division may be used to divide a polynomial by a binomial in the form x r, where r is a constant. 13x 2 5x 12 1x 22 3x 11 x 2 3x 2 5x 1 Answer: 13x 2 6x2 11x x x x 2 13x 2 5x 12 1x 22 Answer: 3x x 2
3 Summary 393 Section 5.4 The greatest common factor (GCF) is the largest factor common to all terms of a polynomial. To factor out the GCF from a polynomial, use the distributive property. A fourterm polynomial may be factored by grouping. Steps to Factor by Grouping 1. Identify and factor out the GCF from all four terms. 2. Factor out the GCF from the first pair of terms. Factor out the GCF from the second pair of terms. (Sometimes it is necessary to factor out the opposite of the GCF.) 3. If the two pairs of terms share a common binomial factor, factor out the binomial factor. Greatest Common Factor and Factoring by Grouping 3x 2 1a b2 6x1a b2 3x1a b2x 3x1a b2122 3x1a b21x 22 60xa 30xb 80ya 40yb 1036xa 3xb 8ya 4yb4 1033x12a b2 4y12a b a b213x 4y2 Section 5.5 Factoring Trinomials ACMethod To factor trinomials of the form ax 2 bx c: 1. Factor out the GCF. 2. Find the product ac. 3. Find two integers whose product is ac and whose sum is b. (If no pair of numbers can be found, then the trinomial is prime.) 4. Rewrite the middle term bx as the sum of two terms whose coefficients are the numbers found in step Factor the polynomial by grouping. 10y 2 35y y 2 7y 42 ac Find two integers whose product is 8 sum is 7. The numbers are 8 and y 2 8y 1y y1y y y 4212y 12 and whose
4 394 Chapter 5 Polynomials TrialandError Method To factor trinomials in the form ax 2 bx c: 1. Factor out the GCF. 2. List the pairs of factors of a and the pairs of factors of c. Consider the reverse order in either list. 3. Construct two binomials of the form Factors of a Factors of c 4. Test each combination of factors until the product of the outer terms and the product of inner terms add to the middle term. 5. If no combination of factors works, the polynomial is prime. The factored form of a perfect square trinomial is the square of a binomial: a 2 2ab b 2 1a b2 2 a 2 2ab b 2 1a b2 2 1 x 21 x 2 10y 2 35y y 2 7y 42 The pairs of factors of 2 are 2 1. The pairs of factors of 4 are y 221y 22 2y 2 2y 4 No 12y 421y 12 2y 2 2y 4 No 12y 121y 42 2y 2 7y 4 No 12y 221y 22 2y 2 2y 4 No 12y 421y 12 2y 2 2y 4 No 12y 121y 42 2y 2 7y 4 Yes Therefore, 10y 2 35y 20 factors as 512y 121y 42. 9w 2 30wz 25z 2 13w w215z2 15z2 2 13w 5z2 2 Section 5.6 Factoring Binomials Factoring Binomials: Summary Difference of squares: a 2 b 2 1a b21a b2 Difference of cubes: a 3 b 3 1a b21a 2 ab b 2 2 Sum of cubes: a 3 b 3 1a b21a 2 ab b u 2 9v 4 15u 3v 2 215u 3v 2 2 8c 3 d 6 12c d 2 214c 2 2cd 2 d w 9 64x 3 13w 3 4x219w 6 12w 3 x 16x 2 2
5 Summary 395 Section 5.7 Additional Factoring Strategies 1. Factor out the GCF (Section 5.4). 2. Identify whether the polynomial has two terms, three terms, or more than three terms. 3. If the polynomial has more than three terms, try factoring by grouping (Section 5.4). 4. If the polynomial has three terms, check first for a perfect square trinomial. Otherwise, factor by using the acmethod or trialanderror method (Section 5.5). 5. If the polynomial has two terms, determine if it fits the pattern for a difference of squares, difference of cubes, or sum of cubes. Remember, a sum of squares is not factorable over the real numbers (Section 5.6). 6. Be sure to factor the polynomial completely. 7. Check by multiplying. 9x 2 4x 9x 3 x19x 4 9x 2 2 x19x 2 9x 42 x13x 4213x 12 4a 2 12ab 9b 2 c 2 4a 2 12ab 9b 2 c 2 12a 3b2 2 c 2 12a 3b c212a 3b c2 Factor out the GCF. Descending order. Factor the trinomial. Group 3 by 1. Perfect square trinomial. Difference of squares. Section 5.8 Solving Equations by Using the Zero Product Rule An equation of the form ax 2 bx c 0, where a 0, is a quadratic equation. The zero product rule states that if a b 0, then a 0 or b 0. The zero product rule can be used to solve a quadratic equation or higherdegree polynomial equation that is factored and equal to zero. 0 x12x 321x 42 x 0 or 2x 3 0 or x 4 0 x 3 2 or x 4 f1x2 ax 2 bx c 1a 02 defines a quadratic function. The xintercepts of a function defined by y f1x2 are determined by finding the real solutions to the equation f1x2 0. The yintercept of a function y f1x2 is at f(0). Find the xintercepts. f1x2 3x 2 8x 5 0 3x 2 8x x 521x 12 3x 5 0 or x 1 0 x 5 3 or x 1 The xintercepts are 1 5 3, 02 and (1, 0). Find the yintercept. f1x2 3x 2 8x 5 f f102 5 The yintercept is 10, 52.
Topic 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 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 informationMath 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2
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
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 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 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 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 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 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 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 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 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 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 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 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 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 informationUnit 7: Factoring Quadratic Polynomials
Unit 7: Factoring Quadratic Polynomials A polynomial is represented by: where the coefficients are real numbers and the exponents are nonnegative integers. Side Note: Examples of real numbers: Examples
More informationOrder of Operations Practice: 1) =
Order of Operations Practice: 1) 2412 3 + 6 = a) 6 b) 42 c) 6 d) 192 2) 36 + 3 3 (1/9)  8 (12) = a) 130 b) 171 c) 183 d) 4,764 1 3) Evaluate: 12 24 2 (  ½ ) + 2 (3) 2 = 4) Evaluate 3y 2 + 8x =, when
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 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 informationMath 3 Variable Manipulation Part 3 Polynomials A
Math 3 Variable Manipulation Part 3 Polynomials A 1 MATH 1 & 2 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does
More informationLecture Guide. Math 90  Intermediate Algebra. Stephen Toner. Intermediate Algebra, 3rd edition. Miller, O'Neill, & Hyde. Victor Valley College
Lecture Guide Math 90  Intermediate Algebra to accompany Intermediate Algebra, 3rd edition Miller, O'Neill, & Hyde Prepared by Stephen Toner Victor Valley College Last updated: 4/17/16 5.1 Exponents &
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 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 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 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 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 informationJust DOS Difference of Perfect Squares. Now the directions say solve or find the real number solutions :
5.4 FACTORING AND SOLVING POLYNOMIAL EQUATIONS To help you with #11 THESE BINOMIALS ARE EITHER GCF, DOS, OR BOTH!!!! Just GCF Just DOS Difference of Perfect Squares Both 1. Break each piece down.. Pull
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 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 informationPolynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.
Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10
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 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 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 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 informationMath 0312 EXAM 2 Review Questions
Name Decide whether the ordered pair is a solution of the given system. 1. 4x + y = 2 2x + 4y = 20 ; (2, 6) Solve the system by graphing. 2. x  y = 6 x + y = 16 Solve the system by substitution. If
More informationPreAlgebra 2. Unit 9. Polynomials Name Period
PreAlgebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (noncomplex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:
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 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 informationLESSON 7.1 FACTORING POLYNOMIALS I
LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I 293 OVERVIEW Here s what you ll learn in this lesson: Greatest Common Factor a. Finding the greatest common factor (GCF) of a set of
More informationSecondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics
Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together
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 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 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 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 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 informationFactoring Trinomials of the Form ax 2 + bx + c, a 1
Factoring Trinomials of the Form ax 2 + bx + c, a 1 When trinomials factor, the resulting terms are binomials. To help establish a procedure for solving these types of equations look at the following patterns.
More informationMath 2 Variable Manipulation Part 3 Polynomials A
Math 2 Variable Manipulation Part 3 Polynomials A 1 MATH 1 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does not
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 informationA quadratic expression is a mathematical expression that can be written in the form 2
118 CHAPTER Algebra.6 FACTORING AND THE QUADRATIC EQUATION Textbook Reference Section 5. CLAST OBJECTIVES Factor a quadratic expression Find the roots of a quadratic equation A quadratic expression is
More informationControlling the Population
Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1
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 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 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 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 informationSolving Quadratic Equations Review
Math III Unit 2: Polynomials Notes 21 Quadratic Equations Solving Quadratic Equations Review Name: Date: Period: Some quadratic equations can be solved by. Others can be solved just by using. ANY quadratic
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 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 informationMath 1 Variable Manipulation Part 6 Polynomials
Name: Math 1 Variable Manipulation Part 6 Polynomials Date: 1 VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does not have
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 informationThe greatest common factor, or GCF, is the largest factor that two or more terms share.
Unit, Lesson Factoring Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product You can factor certain expressions by writing them as the product of
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 informationSolving Linear Equations
Solving Linear Equations Golden Rule of Algebra: Do unto one side of the equal sign as you will do to the other Whatever you do on one side of the equal sign, you MUST do the same exact thing on the other
More informationDay 131 Practice. What Can You Do With Polynomials?
Polynomials Monomial  a Number, a Variable or a PRODUCT of a number and a variable. Monomials cannot have radicals with variables inside, quotients of variables or variables with negative exponents. Degree
More informationPolynomials: Adding, Subtracting, & Multiplying (5.1 & 5.2)
Polynomials: Adding, Subtracting, & Multiplying (5.1 & 5.) Determine if the following functions are polynomials. If so, identify the degree, leading coefficient, and type of polynomial 5 3 1. f ( x) =
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 informationFind two positive factors of 24 whose sum is 10. Make an organized list.
9.5 Study Guide For use with pages 582 589 GOAL Factor trinomials of the form x 2 1 bx 1 c. EXAMPLE 1 Factor when b and c are positive Factor x 2 1 10x 1 24. Find two positive factors of 24 whose sum is
More informationCourse Name: MAT 135 Spring 2017 Master Course Code: N/A. ALEKS Course: Intermediate Algebra Instructor: Master Templates
Course Name: MAT 135 Spring 2017 Master Course Code: N/A ALEKS Course: Intermediate Algebra Instructor: Master Templates Course Dates: Begin: 01/15/2017 End: 05/31/2017 Course Content: 279 Topics (207
More informationWestside. Algebra 2 PreAP
Westside Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for
More informationTHE RING OF POLYNOMIALS. Special Products and Factoring
THE RING OF POLYNOMIALS Special Products and Factoring Special Products and Factoring Upon completion, you should be able to Find special products Factor a polynomial completely Special Products  rules
More informationUnit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any nonnegative power.
MODULE 1 1 Polynomial A function that contains 1 or more or terms. The variables may be to any nonnegative power. 1 Modeling Mathematical modeling is the process of using, and to represent real world
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 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 informationUnit 2: Polynomials Guided Notes
Unit 2: Polynomials Guided Notes Name Period **If found, please return to Mrs. Brandley s room, M 8.** Self Assessment The following are the concepts you should know by the end of Unit 1. Periodically
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 informationWestside Algebra 2 PreAP
Westside Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for
More informationCP Algebra 2 Unit 21: Factoring and Solving Quadratics WORKSHEET PACKET
CP Algebra Unit 1: Factoring and Solving Quadratics WORKSHEET PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor
More informationSolving Quadratic Equations
Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic
More informationSolving Equations Quick Reference
Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number
More informationPOLYNOMIAL EXPRESSIONS PART 1
POLYNOMIAL EXPRESSIONS PART 1 A polynomial is an expression that is a sum of one or more terms. Each term consists of one or more variables multiplied by a coefficient. Coefficients can be negative, so
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 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 informationAlgebra I. Exponents and Polynomials. Name
Algebra I Exponents and Polynomials Name 1 2 UNIT SELFTEST QUESTIONS The Unit Organizer #6 2 LAST UNIT /Experience NAME 4 BIGGER PICTURE DATE Operations with Numbers and Variables 1 CURRENT CURRENT UNIT
More informationAlgebra 2 Summer Work Packet Review and Study Guide
Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the nine specific concepts covered in the
More informationReview 1. 1 Relations and Functions. Review Problems
Review 1 1 Relations and Functions Objectives Relations; represent a relation by coordinate pairs, mappings and equations; functions; evaluate a function; domain and range; operations of functions. Skills
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 informationMAFS Algebra 1. Polynomials. Day 15  Student Packet
MAFS Algebra 1 Polynomials Day 15  Student Packet Day 15: Polynomials MAFS.91.ASSE.1., MAFS.91.ASSE..3a,b, MAFS.91.AAPR..3, MAFS.91.FIF.3.7c I CAN rewrite algebraic expressions in different equivalent
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 informationAre you ready for Algebra 3? Summer Packet *Required for all Algebra 3/Trigonometry Students*
Name: Date: Period: Are you ready for Algebra? Summer Packet *Required for all Students* The course prepares students for Pre Calculus and college math courses. In order to accomplish this, the course
More informationACP ALGEBRA II MIDTERM REVIEW PACKET
ACP ALGEBRA II MIDTERM REVIEW PACKET 08 Name Per Date This review packet includes new problems and a list of problems from the textbook. The answers to the problems from the textbook can be found in the
More informationUnit 2: Polynomials Guided Notes
Unit 2: Polynomials Guided Notes Name Period **If found, please return to Mrs. Brandley s room, M 8.** Self Assessment The following are the concepts you should know by the end of Unit 1. Periodically
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 informationWhich one of the following is the solution to the equation? 1) 4(x  2) + 6 = 2x ) A) x = 5 B) x = 6 C) x = 5 D) x = 6
Review for Final Exam Math 124A (Flatley) Name Which one of the following is the solution to the equation? 1) 4(x  2) + 6 = 2x  14 1) A) x = 5 B) x = 6 C) x = 5 D) x = 6 Solve the linear equation.
More informationSummer 2017 Math Packet
Summer 017 Math Packet for Rising Geometry Students This packet is designed to help you review your Algebra Skills and help you prepare for your Geometry class. Your Geometry teacher will expect you 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 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 information