Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example:
|
|
- Stewart Gordon
- 6 years ago
- Views:
Transcription
1 Polynomials Monomials: 10, 5x, 3x 2, x 3, 4x 2 y 6, or 5xyz 2. A monomial is a product of quantities some of which are unknown. Polynomials: x 3x 2 + x 3, or 4x 2 y 6 + 5xyz 2. A polynomial is a sum of one or more monomials. Mono = one, Poly = many. 1. Factorizations In many problems, it is useful to write polynomials as products. For example, when solving equations: Example: A rectangular house of area 1000m 2 has been enlarged with a corridor or widths 15 m 2 and 10 m 2 like in the picture, resulting in a total area of 1925 m 2. What were the initial width and length of the house? 15 x y 10 (x + 15)(y + 10) = 1925 and xy = 1000 xy + 15y + 10x = 1925 and xy = y + 10x = 775 and xy = x = y and (775 15y)y = 10, 000 x = y and 0 = 15y 2 775y + 10, 000 hence 0 = 3y 2 155y Search (3y )(y ) hence 0 = (3y 80)(y 25) Hence y = 25 and x = = 40 or y = 80 3 and x = = Note: We were lucky to be able to factor 3y 2 155y+2000 = (3y )(y ). We looked among the factors = 2000 at a pair small enough such that 3 + = 155. The pair is too large while is just right. Practice Factorisation by Grouping: a) Solve xy 5x + 7y 35 = 0. 1
2 2 b) Find all integers x, y such that 2xy 3x 18y = 1. c) Factorise x 5 4x 3 y 2 2x 2 + 8y 2. d) Factorise 3x 2 + 2x 8. Solutions: a) xy 5x + 7y 35 = x(y 5) + 7(y 5) = (x + 7)(y 5). Hence x = 7 or y = 5. b) 1 = 2xy 3x 18y = x(2y 3) 9 2y. We need to complete the second term to 9(2y 3) by adding 27. Hence 28 = 2xy 3x 18y + 27 = x(2y 3) 9(2y 3) = (x 9)(2y 3). Keeping in mind that 2y 3 is odd, we can only have 2y 3 = 1, x 9 = 28 hence y = 2, x = 37. 2y 3 = 1, x 9 = 28 hence y = 1, x = 19. 2y 3 = 7, x 9 = 4 hence y = 5, x = 13. 2y 3 = 7, x 9 = 4 hence y = 2, x = 5. c) x 5 4x 3 y 2 2x 2 + 8y 2 = x 3 (x 2 4y 2 ) 2(x 2 4y 2 ) = (x 3 2)(x 2 4y 2 ) = (x 3 2)(x 2y)(x + 2y). d) 3x 2 + 2x 8 = (3x + )(x + ) where = 8 so we may try 2 ( 4), 2 4, 1 ( 8), 1 8. We get 3x 2 + 2x 8 = (3x 4)(x + 2). Grouping and Rearrangement Formulae: Suppose we have two strings of numbers a 1 a 2... a n and b 1 b 2... b n. Among all the sums of products a i b j, where each a is paired with a different b, we want to find the largest sum. a) If a 1 a 2 and b 1 b 2, prove that a 1 b 1 + a 2 b 2 a 1 b 2 + a 2 b 1. b) If a 1 a 2 a 3 and b 1 b 2 b 3, prove that a 1 b 1 + a 2 b 2 + a 3 b 3 a 1 b 2 + a 2 b 3 + a 3 b 1 a 1 b 3 + a 2 b 2 + a 1 b 3, a 1 b 1 + a 2 b 2 + a 3 b 3 a 1 b 3 + a 2 b 1 + a 3 b 2 a 1 b 3 + a 2 b 2 + a 1 b 3. c) In general, if a 1 a 2... a n and b 1 b 2... b n, prove that a 1 b 1 + a 2 b 2 + a 3 b a n b n is the largest among the sums and a 1 b n + a 2 b n a n 1 b 2 + a n b 1 is the smallest among the sums. Solution: a) To compare a 1 b 1 +a 2 b 2 and a 1 b 2 +a 2 b 1, we check the difference: as both factors are 0. a 1 b 1 + a 2 b 2 a 1 b 2 a 2 b 1 = a 1 b 1 a 1 b 2 + a 2 b 2 a 2 b 1 = a 1 (b 1 b 2 ) a 2 (b 1 b 2 ) = (a 1 a 2 )(b 1 b 2 ) 0
3 3 b) Similarly a 1 b 1 + a 2 b 2 + a 3 b 3 a 1 b 2 a 2 b 3 a 3 b 1 = a 1 b 1 a 1 b 2 + a 2 b 2 a 2 b 3 + a 3 b 3 a 3 b 1 = a 1 (b 1 b 2 ) + a 2 (b 2 b 3 ) + a 3 (b 3 b 1 ). Note that (b 1 b 2 ) 0 and (b 2 b 3 ) 0 but (b 3 b 1 ) 0. Moreover, (b 3 b 1 ) = (b 1 b 3 ) = (b 1 b 2 + b 2 b 3 ) = (b 1 b 2 ) (b 2 b 3 ). Hence by grouping again: a 1 b 1 + a 2 b 2 + a 3 b 3 a 1 b 2 a 2 b 3 a 3 b 1 = a 1 (b 1 b 2 ) + a 2 (b 2 b 3 ) + a 3 (b 3 b 1 ) = a 1 (b 1 b 2 ) + a 2 (b 2 b 3 ) a 3 (b 1 b 2 ) a 3 (b 2 b 3 ) = (a 1 a 3 )(b 1 b 2 ) + (a 2 a 3 )(b 2 b 3 ) 0. Similary with the other group. c) Consider a random sum and assume that a 1 b 1 is not in it. Then there are some terms a 1 b j and a k b 1 in it. Then (a 1 b 1 + a k b j ) (a 1 b j + a k b 1 ) = (a 1 a k )(b 1 b j ) 0 so we may increase the sum by replacing a 1 b j + a k b 1 with a 1 b 1 + a k b j. Now we only have (n 1) pairs of numbers which might not be optimally matched, but by applying the previous step for the other pairs a i b i one at a time we obtain the largest sum a 1 b 1 + a 2 b 2 + a 3 b a n b n. The smallest case works out similarly. 2. Factorization Formulae Warm-up: x 2 y 2 = (x y)(x + y) and x 2 + 2xy + y 2 = (x + y) 2. More formulae: Check the formulae below by multiplying through the right-hand-side, using distributivity: n; x n y n = (x y)(x n 1 + x n 2 y xy n 2 + y n 1 ) for positive integers x n + y n = (x y)(x n 1 x n 2 y +... xy n 2 + y n 1 ) for n odd. What can you say about the right-hand-side when n is even? For example: x 2 y 2 = (x y)(x + y); x 3 y 3 = (x y)(x 2 + xy + y 2 ); x 4 y 4 = (x y)(x 3 + x 2 y + xy 2 + y 3 ); x 5 y 5 = (x y)(x 4 + x 3 y + x 2 y 2 + xy 3 + y 4 ); and x 3 + y 3 = (x + y)(x 2 xy + y 2 ); x 5 + y 5 = (x + y)(x 4 x 3 y + x 2 y 2 xy 3 + y 4 ). Example: Factoring polynomials by combining factorization formulae: x 4 y 4 = (x 2 y 2 )(x 2 + y 2 ) = (x y)(x + y)(x 2 + y 2 );
4 4 x 4 + y 4 + x 2 y 2 = x 4 + y 4 + 2x 2 y 2 x 2 y 2 = (x 2 + y 2 ) 2 x 2 y 2 = (x 2 + y 2 xy)(x 2 + y 2 + xy); x 4 + y 4 = x 4 + y 4 + 2x 2 y 2 2x 2 y 2 = (x 2 + y 2 ) 2 2x 2 y 2 = (x 2 + y 2 2xy)(x 2 + y 2 + 2xy); x 6 y 6 = (x 3 y 3 )(x 3 + y 3 ) = (x y)(x 2 + xy + y 2 )(x + y)(x 2 xy + y 2 ). x 7 + x = (x 7 x) + x + x = x(x 6 1) + x 2 + x + 1 = x(x 3 1)(x 3 + 1) + x 2 + x + 1 = x(x 1)(x 2 + x + 1)(x 3 + 1) + x 2 + x + 1 = (x 2 + x + 1)[x(x 1)(x 3 + 1) + 1] The binomial formula: x n + C n 1 x n 1 y + C n 2 x n 2 y C n n 2x 2 y n 2 + C n n 1xy n 1 + y n = (x + y) For example: x 2 + 2xy + y 2 = (x + y) 2 ; x 3 + 3x 2 y + 3xy 2 + y 3 = (x + y) 3 ; x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 = (x + y) 4 ; x 5 + 5x 4 y + 10x 3 y x 2 y 3 + 5xy 4 + y 5 = (x + y) 5 ; x 2 2xy + y 2 = (x y) 2 ; x 3 3x 2 y + 3xy 2 y 3 = (x y) 3 ; x 4 4x 3 y + 6x 2 y 2 4xy 3 + y 4 = (x y) 4 ; x 5 5x 4 y + 10x 3 y 2 10x 2 y 3 + 5xy 4 y 5 = (x y) 5 ; Other factorization formulae: x 3 + y 3 + z 3 3xyz = (x + y + z)(x 2 + y 2 + z 2 xy xz yz). (x + y + z) 2 = x 2 + y 2 + z 2 + 2xy + 2xz + 2yz. (x y 2 1)(x y 2 2) = (x 1 x 2 + y 1 y 2 ) 2 + (x 1 y 2 y 1 x 2 ) Polynomials in one variable x. Completing the square to write a quadratic in vertex form. Consider x 2 + 2xy + y 2 = (x + y) 2. Application:: Writing ax 2 + bx + c = a(x h) 2 + k for some numbers h, k. Example: Example: 3x x + 48 = 3(x x + 16) = 3(x x ) = 3[(x + 5) 2 9]. Why is this useful? a) Solving quadratic equations: 3x x + 48 = 0. Solution:
5 3x x + 48 = 0 becomes 3[(x + 5) 2 9] = 0. Using the difference of two squares this becomes: 3(x + 5 3)(x ) = 0 so 3(x + 2)(x + 8) = 0 so x = 2 or x = 8. b) Find all integers n such that n 2 + 6n + 16 is the square of an integer number. Solution: We are asked to find integer solution for n 2 + 6n + 16 = m 2 But n 2 + 6n + 16 = n n = (n + 3) = m 2 becomes (n + 3) 2 m 2 = 7. Using the difference of squares: (n m)(n + 3 m) = 7 1 = 7 ( 1). We solve simultaneous equations: n m = 7 and n + 3 m = 1. Then 2n + 6 = 6 so n = 6. Or n m = 7 and n + 3 m = 1 so 2n + 6 = 6 so n = Polynomials in one variable x. Division with remainder. Let a(x) and b(x) be two polynomials. Then there exists a unique pair of polynomials q(x) (the quotient) and r(x) (the remainder) for which a(x) = b(x)q(x) + r(x) and degree r(x) < degree q(x). This is similar to the division algorithm for integers. The greatest common divisor d(x) of two polynomials a(x) and b(x) is a polynomial of largest degree which divides exactly into both a(x) and b(x). It can be found by repeated division (Euclid s algorithm). Example: Find the greatest common divisor of x 3 +x 2 3x 6 and x 3 3x 2. By division we get x 3 + x 2 3x 6 = (x 3 3x 2) + (x 2 4) x 3 3x 2 = (x 2 4) x + (x 2) x 2 4 = (x 2) (x + 2) + 0 so x 2 divides x 2 4 and hence also x 3 3x 2 and hence also x 3 +x 2 3x 6. The greatest common divisor is x 2. The Remainder Theorem: Let a be a constant. Dividing a polynomial P (x) by (x a) yields the number P (a) as remainder: Proof: P (x) = (x a)q(x) + P (a).
6 6 Indeed, since (x a) has degree 1, the remainder must have degree 0 and hence be a number, let s call it R. Set x = a to get p(a) = r. P (x) = (x a)q(x) + R. So (x a) is a factor of P (x) if and only if P (a) = 0. Example: Factor P (x) = x 3 5x 2 + 3x + 1. Solution: We search for an integer number a such that P (a) = 0. Then a 3 5a 2 + 3a + 1 = 0 so a 3 5a 2 + 3a = 1. Since the left-hand-side is a multiple of a, then a must be a divisor of 1. We try a = 1. Then P (1) = = 0. The remainder theorem implies that P (x) has x 1 as factor. We can divide P (x) by x 1 as follows: We note that for any number k we have x k (x 1) = x k+1 x k and x k (x 1) = x k+1 + x k. So we can try to write P (x) as a sum of terms of these forms: P (x) = x 3 x 2 4x 2 +4x x+1 = x 2 (x 1) 4x(x 1) (x 1) = (x 1)(x 2 4x 1). By completing the square: P (x) = (x 1)(x 2 4x + 4 5) = (x 1)[(x 2) 2 5] = (x 1)[(x 2) ] = (x 1)(x 2 5)(x 2 5).
1. A polynomial p(x) in one variable x is an algebraic expression in x of the form
POLYNOMIALS Important Points 1. A polynomial p(x) in one variable x is an algebraic expression in x of the form p(x) = a nx n +a n-1x n-1 + a 2x 2 +a 1x 1 +a 0x 0 where a 0, a 1, a 2 a n are constants
More informationUnit 3 Factors & Products
1 Unit 3 Factors & Products General Outcome: Develop algebraic reasoning and number sense. Specific Outcomes: 3.1 Demonstrate an understanding of factors of whole number by determining the: o prime factors
More informationMultiplication 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, 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 informationPOLYNOMIALS. x + 1 x x 4 + x 3. x x 3 x 2. x x 2 + x. x + 1 x 1
POLYNOMIALS A polynomial in x is an expression of the form p(x) = a 0 + a 1 x + a x +. + a n x n Where a 0, a 1, a. a n are real numbers and n is a non-negative integer and a n 0. A polynomial having only
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 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 informationFactorisation CHAPTER Introduction
FACTORISATION 217 Factorisation CHAPTER 14 14.1 Introduction 14.1.1 Factors of natural numbers You will remember what you learnt about factors in Class VI. Let us take a natural number, say 30, and write
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 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 informationMaths Extension 2 - Polynomials. Polynomials
Maths Extension - Polynomials Polynomials! Definitions and properties of polynomials! Factors & Roots! Fields ~ Q Rational ~ R Real ~ C Complex! Finding zeros over the complex field! Factorization & Division
More informationCh 7 Summary - POLYNOMIAL FUNCTIONS
Ch 7 Summary - POLYNOMIAL FUNCTIONS 1. An open-top box is to be made by cutting congruent squares of side length x from the corners of a 8.5- by 11-inch sheet of cardboard and bending up the sides. a)
More informationCollecting Like Terms
MPM1D Unit 2: Algebra Lesson 5 Learning goal: how to simplify algebraic expressions by collecting like terms. Date: Collecting Like Terms WARM-UP Example 1: Simplify each expression using exponent laws.
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 informationL1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen
L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen In this section you will apply the method of long division to divide a polynomial by a binomial. You will also learn to
More informationPOLYNOMIALS. Maths 4 th ESO José Jaime Noguera
POLYNOMIALS Maths 4 th ESO José Jaime Noguera 1 Algebraic expressions Book, page 26 YOUR TURN: exercises 1, 2, 3. Exercise: Find the numerical value of the algebraic expression xy 2 8x + y, knowing that
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 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 informationAlgebraic Expressions
Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly
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 informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 3x 4 2x 3 3x 2 x 7 b. x 1 c. 0.2x 1.x 2 3.2x 3 d. 20 16x 2 20x e. x x 2 x 3 x 4 x f. x 2 6x 2x 6 3x 4 8
More informationL1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen
L1 2.1 Long Division of Polynomials and The Remainder Theorem Lesson MHF4U Jensen In this section you will apply the method of long division to divide a polynomial by a binomial. You will also learn to
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 informationA field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties:
Byte multiplication 1 Field arithmetic A field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties: F is an abelian group under addition, meaning - F is closed under
More informationUnit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power.
MODULE 1 1 Polynomial A function that contains 1 or more or terms. The variables may be to any non-negative power. 1 Modeling Mathematical modeling is the process of using, and to represent real world
More informationP4 Polynomials and P5 Factoring Polynomials
P4 Polynomials and P5 Factoring Polynomials Professor Tim Busken Graduate T.A. Dynamical Systems Program Department of Mathematics San Diego State University June 22, 2011 Professor Tim Busken (Graduate
More informationAdditional Practice Lessons 2.02 and 2.03
Additional Practice Lessons 2.02 and 2.03 1. There are two numbers n that satisfy the following equations. Find both numbers. a. n(n 1) 306 b. n(n 1) 462 c. (n 1)(n) 182 2. The following function is defined
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 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 informationTable of contents. Polynomials Quadratic Functions Polynomials Graphs of Polynomials Polynomial Division Finding Roots of Polynomials
Table of contents Quadratic Functions Graphs of Polynomial Division Finding Roots of Jakayla Robbins & Beth Kelly (UK) Precalculus Notes Fall 2010 1 / 65 Concepts Quadratic Functions The Definition of
More informationAlgebra 2. Factoring Polynomials
Algebra 2 Factoring Polynomials Algebra 2 Bell Ringer Martin-Gay, Developmental Mathematics 2 Algebra 2 Bell Ringer Answer: A Martin-Gay, Developmental Mathematics 3 Daily Learning Target (DLT) Tuesday
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 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 informationIES Parque Lineal - 2º ESO
UNIT5. ALGEBRA Contenido 1. Algebraic expressions.... 1 Worksheet: algebraic expressions.... 2 2. Monomials.... 3 Worksheet: monomials.... 5 3. Polynomials... 6 Worksheet: polynomials... 9 4. Factorising....
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 information( 3) ( ) ( ) ( ) ( ) ( )
81 Instruction: Determining the Possible Rational Roots using the Rational Root Theorem Consider the theorem stated below. Rational Root Theorem: If the rational number b / c, in lowest terms, is a root
More informationAlgebra Review. Terrametra Resources. Lynn Patten
Terrametra Resources Lynn Patten ALGEBRAIC EXPRESSION A combination of ordinary numbers, letter symbols, variables, grouping symbols and operation symbols. Numbers remain fixed in value and are referred
More informationFinal Review Accelerated Advanced Algebra
Name: ate: 1. What are the factors of z + z 2 + 25z + 25? 5. Factor completely: (7x + 2) 2 6 (z + 1)(z + 5)(z 5) (z 1)(z + 5i) 2 (49x + 1)(x 8) (7x 4)(7x + 8) (7x + 4)(7x 8) (7x + 4)(x 9) (z 1)(z + 5i)(z
More informationWarm-Up. Use long division to divide 5 into
Warm-Up Use long division to divide 5 into 3462. 692 5 3462-30 46-45 12-10 2 Warm-Up Use long division to divide 5 into 3462. Divisor 692 5 3462-30 46-45 12-10 2 Quotient Dividend Remainder Warm-Up Use
More informationPermutations and Polynomials Sarah Kitchen February 7, 2006
Permutations and Polynomials Sarah Kitchen February 7, 2006 Suppose you are given the equations x + y + z = a and 1 x + 1 y + 1 z = 1 a, and are asked to prove that one of x,y, and z is equal to a. We
More informationSection 4.3. Polynomial Division; The Remainder Theorem and the Factor Theorem
Section 4.3 Polynomial Division; The Remainder Theorem and the Factor Theorem Polynomial Long Division Let s compute 823 5 : Example of Long Division of Numbers Example of Long Division of Numbers Let
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 information(2) Dividing both sides of the equation in (1) by the divisor, 3, gives: =
Dividing Polynomials Prepared by: Sa diyya Hendrickson Name: Date: Let s begin by recalling the process of long division for numbers. Consider the following fraction: Recall that fractions are just division
More informationCONTENTS COLLEGE ALGEBRA: DR.YOU
1 CONTENTS CONTENTS Textbook UNIT 1 LECTURE 1-1 REVIEW A. p. LECTURE 1- RADICALS A.10 p.9 LECTURE 1- COMPLEX NUMBERS A.7 p.17 LECTURE 1-4 BASIC FACTORS A. p.4 LECTURE 1-5. SOLVING THE EQUATIONS A.6 p.
More informationTEKS: 2A.10F. Terms. Functions Equations Inequalities Linear Domain Factor
POLYNOMIALS UNIT TEKS: A.10F Terms: Functions Equations Inequalities Linear Domain Factor Polynomials Monomial, Like Terms, binomials, leading coefficient, degree of polynomial, standard form, terms, Parent
More informationChapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions
More informationStudent: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed
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 informationChapter 2 Formulas and Definitions:
Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)
More informationComplex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i
Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i 2 = 1 Sometimes we like to think of i = 1 We can treat
More informationFunctions and Equations
Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario Euclid eworkshop # Functions and Equations c 006 CANADIAN
More informationChapter 2 notes from powerpoints
Chapter 2 notes from powerpoints Synthetic division and basic definitions Sections 1 and 2 Definition of a Polynomial Function: Let n be a nonnegative integer and let a n, a n-1,, a 2, a 1, a 0 be real
More informationDividing Polynomials
3-3 3-3 Dividing Polynomials Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Divide using long division. 1. 161 7 2. 12.18 2.1 23 5.8 Divide. 3. 4. 6x + 15y 3 7a 2 ab a 2x + 5y 7a b Objective
More informationUNC Charlotte Super Competition Level 3 Test March 4, 2019 Test with Solutions for Sponsors
. Find the minimum value of the function f (x) x 2 + (A) 6 (B) 3 6 (C) 4 Solution. We have f (x) x 2 + + x 2 + (D) 3 4, which is equivalent to x 0. x 2 + (E) x 2 +, x R. x 2 + 2 (x 2 + ) 2. How many solutions
More information, a 1. , a 2. ,..., a n
CHAPTER Points to Remember :. Let x be a variable, n be a positive integer and a 0, a, a,..., a n be constants. Then n f ( x) a x a x... a x a, is called a polynomial in variable x. n n n 0 POLNOMIALS.
More informationPARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces.
PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION NOAH WHITE The basic aim of this note is to describe how to break rational functions into pieces. For example 2x + 3 1 = 1 + 1 x 1 3 x + 1. The point is that
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 informationHomework 8 Solutions to Selected Problems
Homework 8 Solutions to Selected Problems June 7, 01 1 Chapter 17, Problem Let f(x D[x] and suppose f(x is reducible in D[x]. That is, there exist polynomials g(x and h(x in D[x] such that g(x and h(x
More informationPre-Algebra 2. Unit 9. Polynomials Name Period
Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:
More informationAlgebra I. Book 2. Powered by...
Algebra I Book 2 Powered by... ALGEBRA I Units 4-7 by The Algebra I Development Team ALGEBRA I UNIT 4 POWERS AND POLYNOMIALS......... 1 4.0 Review................ 2 4.1 Properties of Exponents..........
More informationQuestion: 1. Use suitable identities to find the following products:
CH-2 Polynomial Question: 1. Use suitable identities to find the following products: (i) (x + 4) (x + 10) Solution:- (x+4)(x+10) = x 2 +10x+4x+4 x 10 = x 2 +14x+40 (ii) (x + 8) (x 10) Solution: x 2-10x+8x-80
More informationVANA VANI MAT.HR. SEC.SCHOOL Std VIII MATHEMATICS
VANA VANI MAT.HR. SEC.SCHOOL Std VIII MATHEMATICS Holiday assignment 1.Find the common factors of the terms (i) 12x, 36 (ii) 2y, 22xy (iii) 14pq, 28p 2 q 2 (iv) 2x, 3x 2, 4 (v) 6abc, 24ab 2, 12a 2 b (vi)
More informationMATHEMATICS 9 CHAPTER 7 MILLER HIGH SCHOOL MATHEMATICS DEPARTMENT NAME: DATE: BLOCK: TEACHER: Miller High School Mathematics Page 1
MATHEMATICS 9 CHAPTER 7 NAME: DATE: BLOCK: TEACHER: MILLER HIGH SCHOOL MATHEMATICS DEPARTMENT Miller High School Mathematics Page 1 Day 1: Creating expressions with algebra tiles 1. Determine the multiplication
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 informationPolynomial expression
1 Polynomial expression Polynomial expression A expression S(x) in one variable x is an algebraic expression in x term as Where an,an-1,,a,a0 are constant and real numbers and an is not equal to zero Some
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 informationChapter 3: Polynomial and Rational Functions
Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P (x) = a n x n + a n 1 x n 1 + + a 1 x + a 0 The numbers
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 informationSection September 6, If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc.
Section 2.1-2.2 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 informationPolynomials. Henry Liu, 25 November 2004
Introduction Polynomials Henry Liu, 25 November 2004 henryliu@memphis.edu This brief set of notes contains some basic ideas and the most well-known theorems about polynomials. I have not gone into deep
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 informationPARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION. The basic aim of this note is to describe how to break rational functions into pieces.
PARTIAL FRACTIONS AND POLYNOMIAL LONG DIVISION NOAH WHITE The basic aim of this note is to describe how to break rational functions into pieces. For example 2x + 3 = + x 3 x +. The point is that we don
More informationTo Find the Product of Monomials. ax m bx n abx m n. Let s look at an example in which we multiply two monomials. (3x 2 y)(2x 3 y 5 )
5.4 E x a m p l e 1 362SECTION 5.4 OBJECTIVES 1. Find the product of a monomial and a polynomial 2. Find the product of two polynomials 3. Square a polynomial 4. Find the product of two binomials that
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 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 information3.5. Dividing Polynomials. LEARN ABOUT the Math. Selecting a strategy to divide a polynomial by a binomial
3.5 Dividing Polynomials GOAL Use a variety of strategies to determine the quotient when one polynomial is divided by another polynomial. LEARN ABOU the Math Recall that long division can be used to determine
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 informationAlgebra III Chapter 2 Note Packet. Section 2.1: Polynomial Functions
Algebra III Chapter 2 Note Packet Name Essential Question: Section 2.1: Polynomial Functions Polynomials -Have nonnegative exponents -Variables ONLY in -General Form n ax + a x +... + ax + ax+ a n n 1
More informationClass IX Chapter 2 Polynomials Maths
NCRTSOLUTIONS.BLOGSPOT.COM Class IX Chapter 2 Polynomials Maths Exercise 2.1 Question 1: Which of the following expressions are polynomials in one variable and which are No. It can be observed that the
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 informationUNIT 5 VOCABULARY: POLYNOMIALS
3º ESO Bilingüe Page 1 UNIT 5 VOCABULARY: POLYNOMIALS 1.1. Monomials A monomial is an algebraic expression consisting of only one term. A monomial can be any of the following: A constant: 2 4-5 A variable:
More information6x 3 12x 2 7x 2 +16x 7x 2 +14x 2x 4
2.3 Real Zeros of Polynomial Functions Name: Pre-calculus. Date: Block: 1. Long Division of Polynomials. We have factored polynomials of degree 2 and some specific types of polynomials of degree 3 using
More informationName: Chapter 7: Exponents and Polynomials
Name: Chapter 7: Exponents and Polynomials 7-1: Integer Exponents Objectives: Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents. You
More informationA polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers.
LEAVING CERT Honours Maths notes on Algebra. A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. The degree is the highest power of x. 3x 2 + 2x
More informationDownloaded from
Question 1: Exercise 2.1 The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) Page 1 of 24 (iv) (v) (v) Page
More informationLagrange s polynomial
Lagrange s polynomial Nguyen Trung Tuan November 16, 2016 Abstract In this article, I will use Lagrange polynomial to solve some problems from Mathematical Olympiads. Contents 1 Lagrange s interpolation
More informationMath 10 - Unit 5 Final Review - Polynomials
Class: Date: Math 10 - Unit 5 Final Review - Polynomials Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Factor the binomial 44a + 99a 2. a. a(44 + 99a)
More informationA DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any
Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»
More informationWarm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 2
Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Factor each expression. 1. 3x 6y 2. a 2 b 2 3(x 2y) (a + b)(a b) Find each product. 3. (x 1)(x + 3) 4. (a + 1)(a 2 + 1) x 2 + 2x 3 a 3 + a 2 +
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 informationCP Algebra 2 Unit 2-1: 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 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 informationANSWERS. CLASS: VIII TERM - 1 SUBJECT: Mathematics. Exercise: 1(A) Exercise: 1(B)
ANSWERS CLASS: VIII TERM - 1 SUBJECT: Mathematics TOPIC: 1. Rational Numbers Exercise: 1(A) 1. Fill in the blanks: (i) -21/24 (ii) -4/7 < -4/11 (iii)16/19 (iv)11/13 and -11/13 (v) 0 2. Answer True or False:
More informationAlg 1B Chapter 7 Final Exam Review
Name: Class: Date: ID: A Alg B Chapter 7 Final Exam Review Please answer all questions and show your work. Simplify ( 2) 4. 2. Simplify ( 4) 4. 3. Simplify 5 2. 4. Simplify 9x0 y 3 z 8. 5. Simplify 7w0
More informationI CAN classify polynomials by degree and by the number of terms.
13-1 Polynomials I CAN classify polynomials by degree and by the number of terms. 13-1 Polynomials Insert Lesson Title Here Vocabulary monomial polynomial binomial trinomial degree of a polynomial 13-1
More informationPower and Polynomial Functions. College Algebra
Power and Polynomial Functions College Algebra Power Function A power function is a function that can be represented in the form f x = kx % where k and p are real numbers, and k is known as the coefficient.
More informationSYMBOL NAME DESCRIPTION EXAMPLES. called positive integers) negatives, and 0. represented as a b, where
EXERCISE A-1 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 informationCM2104: Computational Mathematics General Maths: 2. Algebra - Factorisation
CM204: Computational Mathematics General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of simplifying algebraic expressions.
More informationCHAPTER 1 POLYNOMIALS
1 CHAPTER 1 POLYNOMIALS 1.1 Removing Nested Symbols of Grouping Simplify. 1. 4x + 3( x ) + 4( x + 1). ( ) 3x + 4 5 x 3 + x 3. 3 5( y 4) + 6 y ( y + 3) 4. 3 n ( n + 5) 4 ( n + 8) 5. ( x + 5) x + 3( x 6)
More information