A.4 Polynomial Division; Synthetic Division


 June Short
 2 years ago
 Views:
Transcription
1 SECTION A.4 Polynomial Division; Synthetic Division 977 A.4 Polynomial Division; Synthetic Division OBJECTIVES 1 Divide Polynomials Using Long Division 2 Divide Polynomials Using Synthetic Division 1 Divide Polynomials Using Long Division The procedure for dividing two polynomials is similar to the procedure for dividing two integers. EXAMPLE 1 Dividing Two Integers Divide 842 by 15. Solution Divisor : ; Quotient ; Dividend ; 5 # 15 (Subtract) ; 6 # 15 (Subtract) ; Remainder So, = In the long division process detailed in Example 1, the number 15 is called the divisor, the number 842 is called the dividend, the number 56 is called the quotient, and the number 2 is called the remainder. To check the answer obtained in a division problem, multiply the quotient by the divisor and add the remainder. The answer should be the dividend. 1Quotient21Divisor2 + Remainder = Dividend For example, we can check the results obtained in Example 1 as follows: = = 842 To divide two polynomials, we first must write each polynomial in standard form. The process then follows a pattern similar to that of Example 1. The next example illustrates the procedure. EXAMPLE 2 Dividing Two Polynomials Find the quotient and the remainder when 3x 3 + 4x 2 + x + 7 is divided by x Solution Each polynomial is in standard form. The dividend is 3x 3 + 4x 2 + x + 7, and the divisor is x
2 978 APPENDIX Review STEP 1: Divide the leading term of the dividend, 3x 3, by the leading term of the divisor, x 2. Enter the result, 3x, over the term 3x 3, as follows: 3x x x 3 + 4x 2 + x + 7 STEP 2: Multiply 3x by x and enter the result below the dividend. 3x x x 3 + 4x 2 + x + 7 3x 3 + 3x ; 3x # (x 2 + 1) = 3x 3 + 3x q Notice that we align the 3x term under the x to make the next step easier. STEP 3: Subtract and bring down the remaining terms. 3x x x 3 + 4x 2 + x + 7 3x 3 + 3x 4x 22x + 7 ; Subtract (change the signs and add) ; Bring down the 4x 2 and the 7. STEP 4: Repeat Steps 1 3 using 4x 22x + 7 as the dividend. 3x + 4 x x 3 + 4x 2 + x + 7 3x 3 + 3x 4x 22x + 7 4x x + 3 ; ; Divide 4x 2 by x 2 to get 4. ; Multiply x by 4; subtract. Since does not divide 2x evenly (that is, the result is not a monomial), the process ends.the quotient is 3x + 4, and the remainder is 2x + 3. x 2 CHECK: (Quotient)(Divisor) + Remainder = 13x + 421x x + 32 = 3x 3 + 4x 2 + 3x x + 32 = 3x 3 + 4x 2 + x + 7 = Dividend Then 3x 3 + 4x 2 + x + 7 x = 3x x + 3 x The next example combines the steps involved in long division. EXAMPLE 3 Dividing Two Polynomials Find the quotient and the remainder when x 43x 3 + 2x  5 is divided by x 2  x + 1
3 SECTION A.4 Polynomial Division; Synthetic Division 979 Solution In setting up this division problem, it is necessary to leave a space for the missing x 2 term in the dividend. Divisor : Subtract : Subtract : Subtract : x 22x  3 x 2  x + 1x 43x 3 + 2x  5 x 4  x 3 + x 22x 3  x 2 + 2x  52x 3 + 2x 22x 3x 2 + 4x  53x 2 + 3x  3 x  2 ; Quotient ; Dividend ; Remainder CHECK: (Quotient)(Divisor) + Remainder = 1x 22x  321x 2  x x  2 = x 4  x 3 + x 22x 3 + 2x 22x  3x 2 + 3x x  2 = x 43x 3 + 2x  5 = Dividend As a result, x 43x 3 + 2x  5 x 2 = x 2 x  22x x + 1 x 2  x + 1 The process of dividing two polynomials leads to the following result: Theorem Let Q be a polynomial of positive degree and let P be a polynomial whose degree is greater than the degree of Q. The remainder after dividing P by Q is either the zero polynomial or a polynomial whose degree is less than the degree of the divisor Q. 2 NOW WORK PROBLEM 9. Divide Polynomials Using Synthetic Division To find the quotient as well as the remainder when a polynomial of degree 1 or higher is divided by x  c, a shortened version of long division, called synthetic division, makes the task simpler. To see how synthetic division works, we will use long division to divide the polynomial 2x 3  x by x x 2 + 5x + 15 x  32x 3  x x 36x 2 # 5x x 215x 15x x CHECK: (Divisor) (Quotient) + Remainder ; Quotient ; Remainder = 1x x 2 + 5x = 2x 3 + 5x x  6x 215x = 2x 3  x 2 + 3
4 980 APPENDIX Review The process of synthetic division arises from rewriting long division in a more compact form, using simpler notation. For example, in the long division on p. 979, the terms in blue are not really necessary because they are identical to the terms directly above them. With these terms removed, we have 2x 2 + 5x + 15 x  32x 3  x x 2 5x 215x 15x Most of the x s that appear in this process can also be removed, provided that we are careful about positioning each coefficient. In this regard, we will need to use 0 as the coefficient of x in the dividend, because that power of x is missing. Now we have 2x 2 + 5x + 15 x We can make this display more compact by moving the lines up until the numbers in color align horizontally. 2x 2 + 5x + 15 x ~ Row 4 Because the leading coefficient of the divisor is always 1, we know that the leading coefficient of the dividend will also be the leading coefficient of the quotient. So we place the leading coefficient of the quotient, 2, in the circled position. Now, the first three numbers in row 4 are precisely the coefficients of the quotient, and the last number in row 4 is the remainder. Thus, row 1 is not really needed, so we can compress the process to three rows, where the bottom row contains both the coefficients of the quotient and the remainder. x (subtract) Recall that the entries in row 3 are obtained by subtracting the entries in row 2 from those in row 1. Rather than subtracting the entries in row 2, we can
5 SECTION A.4 Polynomial Division; Synthetic Division 981 change the sign of each entry and add.with this modification, our display will look like this: x (add) Notice that the entries in row 2 are three times the prior entries in row 3. Our last modification to the display replaces the x  3 by 3. The entries in row 3 give the quotient and the remainder, as shown next (add) Quotient Remainder 2x 2 5x Let s go through an example step by step. EXAMPLE 4 Using Synthetic Division to Find the Quotient and Remainder Use synthetic division to find the quotient and remainder when x 34x 25 is divided by x  3 Solution STEP 1: Write the dividend in descending powers of x. Then copy the coefficients, remembering to insert a 0 for any missing powers of x STEP 2: Insert the usual division symbol. In synthetic division, the divisor is of the form x  c, and c is the number placed to the left of the division symbol. Here, since the divisor is x  3, we insert 3 to the left of the division symbol STEP 3: Bring the 1 down two rows, and enter it in row p 1 STEP 4: Multiply the latest entry in row 3 by 3, and place the result in row 2, one column over to the right STEP 5: Add the entry in row 2 to the entry above it in row 1, and enter the sum in row
6 982 APPENDIX Review STEP 6: Repeat Steps 4 and 5 until no more entries are available in row STEP 7: The final entry in row 3, the 14, is the remainder; the other entries in row 3, the 1, 1, and 3, are the coefficients (in descending order) of a polynomial whose degree is 1 less than that of the dividend. This is the quotient. Thus, Quotient = x 2  x  3 Remainder = 14 CHECK: (Divisor)(Quotient) + Remainder = 1x  321x 2  x = 1x 3  x 23x  3x 2 + 3x = x 34x 25 = Dividend Let s do an example in which all seven steps are combined. EXAMPLE 5 Using Synthetic Division to Verify a Factor Use synthetic division to show that x + 3 is a factor of 2x 5 + 5x 42x 3 + 2x 22x + 3 Solution The divisor is x + 3 = x , so we place 3 to the left of the division symbol. Then the row 3 entries will be multiplied by 3, entered in row 2, and added to row Because the remainder is 0, we have 1Divisor21Quotient2 + Remainder = 1x x 4  x 3 + x 2  x + 12 = 2x 5 + 5x 42x 3 + 2x 22x + 3 As we see, x + 3 is a factor of 2x 5 + 5x 42x 3 + 2x 22x + 3. As Example 5 illustrates, the remainder after division gives information about whether the divisor is, or is not, a factor. NOW WORK PROBLEMS 23 AND 33.
7 982 APPENDIX Review A.4 Assess Your Understanding Concepts and Vocabulary 1. To check division, the expression that is being divided, the dividend, should equal the product of the and the plus the. 2. To divide 2x 35x + 1 by x + 3 using synthetic division, the first step is to write. 3. True or False: In using synthetic division, the divisor is always a polynomial of degree 1, whose leading coefficient is 1.
8 SECTION A.4 Polynomial Division; Synthetic Division True or False: means x 3 + 3x 2 + 2x + 1 x + 2 = 5x 27x x + 2. Skill Building In Problems 5 20, find the quotient and the remainder. Check your work by verifying that 1Quotient21Divisor2 + Remainder = Dividend 5. 4x 33x 2 + x + 1 divided by x x 33x 2 + x + 1 divided by x x 43x 2 + x + 1 divided by x x 53x 2 + x + 1 divided by 2x x 43x 3 + x + 1 divided by 2x 2 + x x 3 + x 24 divided by x x 2 + x 4 divided by x 2 + x x 3  x 2 + x  2 divided by x + 2 3x 3  x 2 + x  2 divided by x 2 5x 4  x 2 + x  2 divided by x x 5  x 2 + x  2 divided by 3x 31 3x 4  x 3 + x  2 divided by 3x 2 + x + 13x 42x  1 divided by x x 2 + x 4 divided by x 2  x x 3  a 3 divided by x  a 20. x 5  a 5 divided by x  a In Problems 21 32, use synthetic division to find the quotient and remainder. 21. x 3  x 2 + 2x + 4 divided by x x 3 + 2x 23x + 1 divided by x x 3 + 2x 2  x + 3 divided by x x 54x 3 + x divided by x x 63x 4 + x divided by x x x divided by x x 51 divided by x x 3 + 2x 2  x + 1 divided by x + 2 x 4 + x divided by x  2 x 5 + 5x 310 divided by x x divided by x x divided by x + 1 In Problems 33 42, use synthetic division to determine whether x  c is a factor of the given polynomial x 33x 28x + 4; x x 3 + 5x 2 + 8; x x 46x 35x + 10; x x x ; x x 664x 4 + x 215; x x 415x 24; x  2 2x 618x 4 + x 29; x + 3 x 616x 4 + x 216; x x 4  x 3 + 2x  1; x x 4 + x 33x + 1; x Find the sum of a, b, c, and d if Discussion and Writing x 32x 2 + 3x + 5 x + 2 = ax 2 + bx + c + d x When dividing a polynomial by x  c, do you prefer to use long division or synthetic division? Does the value of c make a difference to you in choosing? Give reasons.
Dividing Polynomials; Remainder and Factor Theorems
Dividing Polynomials; Remainder and Factor Theorems In this section we will learn how to divide polynomials, an important tool needed in factoring them. This will begin our algebraic study of polynomials.
More informationSect 3.2 Synthetic Division
94 Objective 1: Sect 3.2 Synthetic Division Division Algorithm Recall that when dividing two numbers, we can check our answer by the whole number (quotient) times the divisor plus the remainder. This should
More informationMonomials with the same variables to the same powers are called like terms, If monomials are like terms only their coefficients can differ.
Chapter 7.1 Introduction to Polynomials A monomial is an expression that is a number, a variable or the product of a number and one or more variables with nonnegative exponents. Monomials that are real
More informationDIVISION OF POLYNOMIALS
5.5 Division of Polynomials (533) 89 5.5 DIVISION OF POLYNOMIALS In this section Dividing a Polynomial by a Monomial Dividing a Polynomial by a Binomial Synthetic Division Division and Factoring We began
More informationDividing Polynomials
4.3 Dividing Polynomials Essential Question Essential Question How can you use the factors of a cubic polynomial to solve a division problem involving the polynomial? Dividing Polynomials Work with a partner.
More informationDividing Polynomials VOCABULARY
 Dividing Polynomials TEKS FOCUS TEKS ()(C) Determine the quotient of a polynomial of degree three and degree four when divided by a polynomial of degree one and of degree two. TEKS ()(A) Apply mathematics
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 informationDecimals Adding and Subtracting
1 Decimals Adding and Subtracting Decimals are a group of digits, which express numbers or measurements in units, tens, and multiples of 10. The digits for units and multiples of 10 are followed by a decimal
More informationSYNTHETIC DIVISION AND THE FACTOR THEOREM
628 (11 48) Chapter 11 Functions In this section Synthetic Division The Factor Theorem Solving Polynomial Equations 11.6 SYNTHETIC DIVISION AND THE FACTOR THEOREM In this section we study functions defined
More informationCLASS NOTES. We bring down (copy) the leading coefficient below the line in the same column.
SYNTHETIC DIVISION CLASS NOTES When factoring or evaluating polynomials we often find that it is convenient to divide a polynomial by a linear (first degree) binomial of the form x k where k is a real
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 information23 The Remainder and Factor Theorems
Factor each polynomial completely using the given factor and long division. 1. x 3 + 2x 2 23x 60; x + 4 So, x 3 + 2x 2 23x 60 = (x + 4)(x 2 2x 15). Factoring the quadratic expression yields x 3 + 2x 2
More informationName: where Nx ( ) and Dx ( ) are the numerator and
Oblique and Nonlinear Asymptote Activity Name: Prior Learning Reminder: Rational Functions In the past we discussed vertical and horizontal asymptotes of the graph of a rational function of the form m
More informationOperations on Decimals
Operations on Decimals Addition and subtraction of decimals To add decimals, write the numbers so that the decimal points are on a vertical line. Add as you would with whole numbers. Then write the decimal
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationYOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!
DETAILED SOLUTIONS AND CONCEPTS  DECIMALS AND WHOLE NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! YOU MUST
More informationChanging a Mixed Number to an Improper Fraction
Example: Write 48 4 48 4 = 48 8 4 8 = 8 8 = 2 8 2 = 4 in lowest terms. Find a number that divides evenly into both the numerator and denominator of the fraction. For the fraction on the left, there are
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More information3.2. The Remainder Theorem. Investigate Polynomial Division
3.2 The Remainder Theorem Focus on... describing the relationship between polynomial long division and synthetic division dividing polynomials by binomials of the form x  a using long division or synthetic
More informationParamedic Program PreAdmission Mathematics Test Study Guide
Paramedic Program PreAdmission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page
More information2.4 Real Zeros of Polynomial Functions
SECTION 2.4 Real Zeros of Polynomial Functions 197 What you ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower
More informationPolynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c
More information3.2 The Factor Theorem and The Remainder Theorem
3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial
More informationImproper Fractions and Mixed Numbers
This assignment includes practice problems covering a variety of mathematical concepts. Do NOT use a calculator in this assignment. The assignment will be collected on the first full day of class. All
More informationIn this lesson you will learn to find zeros of polynomial functions that are not factorable.
2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has
More informationCHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS
CHAPTER 2: POLYNOMIAL AND RATIONAL FUNCTIONS 2.01 SECTION 2.1: QUADRATIC FUNCTIONS (AND PARABOLAS) PART A: BASICS If a, b, and c are real numbers, then the graph of f x = ax2 + bx + c is a parabola, provided
More informationChapter 4 Fractions and Mixed Numbers
Chapter 4 Fractions and Mixed Numbers 4.1 Introduction to Fractions and Mixed Numbers Parts of a Fraction Whole numbers are used to count whole things. To refer to a part of a whole, fractions are used.
More information3. Power of a Product: Separate letters, distribute to the exponents and the bases
Chapter 5 : Polynomials and Polynomial Functions 5.1 Properties of Exponents Rules: 1. Product of Powers: Add the exponents, base stays the same 2. Power of Power: Multiply exponents, bases stay the same
More informationFACTORING POLYNOMIALS
296 (540) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated
More informationAn Insight into Division Algorithm, Remainder and Factor Theorem
An Insight into Division Algorithm, Remainder and Factor Theorem Division Algorithm Recall division of a positive integer by another positive integer For eample, 78 7, we get and remainder We confine the
More informationRecall the process used for adding decimal numbers. 1. Place the numbers to be added in vertical format, aligning the decimal points.
2 MODULE 4. DECIMALS 4a Decimal Arithmetic Adding Decimals Recall the process used for adding decimal numbers. Adding Decimals. To add decimal numbers, proceed as follows: 1. Place the numbers to be added
More informationReteaching. Properties of Operations
 Properties of Operations The commutative properties state that changing the order of addends or factors in a multiplication or addition expression does not change the sum or the product. Examples: 5
More informationDividing Polynomials, The Remainder Theorem and Factor Theorem
College Algebra  MAT 161 Page: 1 Copyright 2009 Killoran Dividing Polynomials, The Remainder Theorem and Factor Theorem 1 Long Division: a b D c C r b a is the Quotient b is the Divisor c is the Dividend
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 informationCourse notes on Number Theory
Course notes on Number Theory In Number Theory, we make the decision to work entirely with whole numbers. There are many reasons for this besides just mathematical interest, not the least of which is that
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationUNIT 3: POLYNOMIALS AND ALGEBRAIC FRACTIONS. A polynomial is an algebraic expression that consists of a sum of several monomials. x n 1...
UNIT 3: POLYNOMIALS AND ALGEBRAIC FRACTIONS. Polynomials: A polynomial is an algebraic expression that consists of a sum of several monomials. Remember that a monomial is an algebraic expression as ax
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 informationFACTORING OUT COMMON FACTORS
278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More 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 information2.5 Zeros of a Polynomial Functions
.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the xaxis and
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationSection R.2. Fractions
Section R.2 Fractions Learning objectives Fraction properties of 0 and 1 Writing equivalent fractions Writing fractions in simplest form Multiplying and dividing fractions Adding and subtracting fractions
More informationeday Lessons Mathematics Grade 8 Student Name:
eday Lessons Mathematics Grade 8 Student Name: Common Core State Standards Expressions and Equations Work with radicals and integer exponents. 3. Use numbers expressed in the form of a single digit times
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 information(2 4 + 9)+( 7 4) + 4 + 2
5.2 Polynomial Operations At times we ll need to perform operations with polynomials. At this level we ll just be adding, subtracting, or multiplying polynomials. Dividing polynomials will happen in future
More informationPreviously, you learned the names of the parts of a multiplication problem. 1. a. 6 2 = 12 6 and 2 are the. b. 12 is the
Tallahassee Community College 13 PRIME NUMBERS AND FACTORING (Use your math book with this lab) I. Divisors and Factors of a Number Previously, you learned the names of the parts of a multiplication problem.
More informationWentzville School District Algebra 1: Unit 8 Stage 1 Desired Results
Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Unit Title: Quadratic Expressions & Equations Course: Algebra I Unit 8  Quadratic Expressions & Equations Brief Summary of Unit: At
More informationThe Euclidean Algorithm
The Euclidean Algorithm A METHOD FOR FINDING THE GREATEST COMMON DIVISOR FOR TWO LARGE NUMBERS To be successful using this method you have got to know how to divide. If this is something that you have
More informationPolynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
More informationMath Rational Functions
Rational Functions Math 3 Rational Functions A rational function is the algebraic equivalent of a rational number. Recall that a rational number is one that can be epressed as a ratio of integers: p/q.
More informationNow that we have a handle on the integers, we will turn our attention to other types of numbers.
1.2 Rational Numbers Now that we have a handle on the integers, we will turn our attention to other types of numbers. We start with the following definitions. Definition: Rational Number any number that
More information5.1 The Remainder and Factor Theorems; Synthetic Division
5.1 The Remainder and Factor Theorems; Synthetic Division In this section you will learn to: understand the definition of a zero of a polynomial function use long and synthetic division to divide polynomials
More informationApplication. Outline. 31 Polynomial Functions 32 Finding Rational Zeros of. Polynomial. 33 Approximating Real Zeros of.
Polynomial and Rational Functions Outline 31 Polynomial Functions 32 Finding Rational Zeros of Polynomials 33 Approximating Real Zeros of Polynomials 34 Rational Functions Chapter 3 Group Activity:
More informationPREPARATION FOR MATH TESTING at CityLab Academy
PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRETEST
More informationMathematics of Cryptography
CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter
More informationAccuplacer Arithmetic Study Guide
Testing Center Student Success Center Accuplacer Arithmetic Study Guide I. Terms Numerator: which tells how many parts you have (the number on top) Denominator: which tells how many parts in the whole
More informationA Short Introduction to Binary Numbers
A Short Introduction to Binary Numbers Brian J. Shelburne Department of Mathematics and Computer Science Wittenberg University 0. Introduction The development of the computer was driven by the need to
More information63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15.
9.4 (927) 517 Gear ratio d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases? decreases 100 80 60 40 20 27in. wheel, 44 teeth
More 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 informationMATH Fundamental Mathematics II.
MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/funmath2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics
More informationChapter 3. Algebra. 3.1 Rational expressions BAa1: Reduce to lowest terms
Contents 3 Algebra 3 3.1 Rational expressions................................ 3 3.1.1 BAa1: Reduce to lowest terms...................... 3 3.1. BAa: Add, subtract, multiply, and divide............... 5
More informationWritten methods for division of whole numbers
Written methods for division of whole numbers The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method
More informationUnit 1: Polynomials. Expressions:  mathematical sentences with no equal sign. Example: 3x + 2
Pure Math 0 Notes Unit : Polynomials Unit : Polynomials : Reviewing Polynomials Epressions:  mathematical sentences with no equal sign. Eample: Equations:  mathematical sentences that are equated with
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 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 informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationSOLVING POLYNOMIAL EQUATIONS
C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra
More informationFOIL FACTORING. Factoring is merely undoing the FOIL method. Let s look at an example: Take the polynomial x²+4x+4.
FOIL FACTORING Factoring is merely undoing the FOIL method. Let s look at an example: Take the polynomial x²+4x+4. First we take the 3 rd term (in this case 4) and find the factors of it. 4=1x4 4=2x2 Now
More informationGrade 9 Mathematics Unit #1 Number Sense SubUnit #1 Rational Numbers. with Integers Divide Integers
Page1 Grade 9 Mathematics Unit #1 Number Sense SubUnit #1 Rational Numbers Lesson Topic I Can 1 Ordering & Adding Create a number line to order integers Integers Identify integers Add integers 2 Subtracting
More informationCommon Core State Standards for Math Grades K  7 2012
correlated to the Grades K  7 The Common Core State Standards recommend more focused and coherent content that will provide the time for students to discuss, reason with, reflect upon, and practice more
More informationSometimes it is easier to leave a number written as an exponent. For example, it is much easier to write
4.0 Exponent Property Review First let s start with a review of what exponents are. Recall that 3 means taking four 3 s and multiplying them together. So we know that 3 3 3 3 381. You might also recall
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 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 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 informationMore Zeroes of Polynomials. Elementary Functions. The Rational Root Test. The Rational Root Test
More Zeroes of Polynomials In this lecture we look more carefully at zeroes of polynomials. (Recall: a zero of a polynomial is sometimes called a root.) Our goal in the next few presentations is to set
More informationWhy Vedic Mathematics?
Why Vedic Mathematics? Many Indian Secondary School students consider Mathematics a very difficult subject. Some students encounter difficulty with basic arithmetical operations. Some students feel it
More information3.7 Complex Zeros; Fundamental Theorem of Algebra
SECTION.7 Complex Zeros; Fundamental Theorem of Algebra 2.7 Complex Zeros; Fundamental Theorem of Algebra PREPARING FOR THIS SECTION Before getting started, review the following: Complex Numbers (Appendix,
More informationLagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.
Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method
More informationGrade 6 Math Circles October 25 & 26, Number Systems and Bases
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles October 25 & 26, 2016 Number Systems and Bases Numbers are very important. Numbers
More informationMultiplication with Whole Numbers
Math 952 1.3 "Multiplication and Division with Whole Numbers" Objectives * Be able to multiply and understand the terms factor and product. * Properties of multiplication commutative, associative, distributive,
More informationThe Crescent Primary School Calculation Policy
The Crescent Primary School Calculation Policy Examples of calculation methods for each year group and the progression between each method. January 2015 Our Calculation Policy This calculation policy has
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 informationAddition and Multiplication of Polynomials
LESSON 0 addition and multiplication of polynomials LESSON 0 Addition and Multiplication of Polynomials Base 0 and Base  Recall the factors of each of the pieces in base 0. The unit block (green) is x.
More informationPolynomials. Solving Equations by Using the Zero Product Rule
mil23264_ch05_303396 9:21:05 06:16 PM Page 303 Polynomials 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials 5.4 Greatest
More informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
More informationWritten methods for addition of whole numbers
Written methods for addition of whole numbers The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationWhole Numbers. hundred ten one
Whole Numbers WHOLE NUMBERS: WRITING, ROUNDING The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The natural numbers (counting numbers) are 1, 2, 3, 4, 5, and so on. The whole numbers are 0, 1, 2, 3, 4,
More informationPolynomials Classwork
Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th degree polynomial function Def.: A polynomial function of degree n in the vaiable
More informationPolynomial Equations and Factoring
7 Polynomial Equations and Factoring 7.1 Adding and Subtracting Polynomials 7.2 Multiplying Polynomials 7.3 Special Products of Polynomials 7.4 Dividing Polynomials 7.5 Solving Polynomial Equations in
More informationPlacement Test Review Materials for
Placement Test Review Materials for 1 To The Student This workbook will provide a review of some of the skills tested on the COMPASS placement test. Skills covered in this workbook will be used on the
More informationHOSPITALITY Math Assessment Preparation Guide. Introduction Operations with Whole Numbers Operations with Integers 9
HOSPITALITY Math Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre at George
More informationMathematics. GSE Algebra II/Advanced Algebra Unit 3: Polynomial Functions
Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Algebra II/Advanced Algebra Unit 3: Polynomial Functions These materials are for nonprofit educational purposes only. Any other use
More information1 Gaussian Elimination
Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 GaussJordan reduction and the Reduced
More information3.6 The Real Zeros of a Polynomial Function
SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,
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 information