7 Quadratic Expressions
|
|
- Camilla Higgins
- 7 years ago
- Views:
Transcription
1 7 Quadratic Expressions A quadratic expression (Latin quadratus squared ) is an expression involving a squared term, e.g., x + 1, or a product term, e.g., xy x + 1. (A linear expression such as x + 1 is obviously non-quadratic.) 7.1 Expansion (Multiplication) We have seen in Topic, Section how to multiply, or expand, simple expressions involving one linear factor such as (x 5) i.e. (x 5) = (x) 5 = 6x 15. Now we wish to multiply (expand) two linear expressions - the product is naturally a quadratic expression. Always simplify the quadratic expression, if possible. Example: (x 5)(x + ) = x(x + ) 5(x + ) = x x + x 5x 10 = x + 4x 5x 10 = x x 10 on simplification. The order of the factors is unimportant, i.e. (x 5)(x + ) = (x + )(x 5) 7 1
2 Exercises 7.1: Multiply (expand) (i) (x + )(x 4) (ii) (x )(x ) (iii) (x 1)( x) (iv) (x + 1)(y + ) (v) (x ) (vi) (x + 1)(x 1) (vii) (7x + 4)(7x 4) (viii) (x + a)(x + b) (ix) (x + a)(x a) (x) (x a) (xi) (x y)(x + 4y) (xii) (x 5t) Note: In (ix), the factors x + a, x a differ only in the sign in front of a, leading to the difference of squares x a. (The term in ax is +ax ax = 0, i.e., there is no term in ax.) In (x), the square of x a, namely (x a)(x a), will have a term in ax, i.e., ax ax(= ax). The squared terms are x x = x and ( a)( a) = +a. 7. Factorizing In Section 1 above, we were given the factors and we had to find the equivalent unfactored expression. Now we are given an unfactored expression and we are asked to find the factors, i.e., to factorize the expression. Guesswork, to some extent, is involved in this technique, until you become proficient at factorizing. 7
3 Example: Factorize x x 1 Solution: we look for an answer of the kind (x + a)(x + b) [= x + (a + b)x + ab by Exercise 7.1, (viii) above] This means a + b = 1, ab = 1 in our Example. Solving by trial-and-error gives a =, b = 4 (or a = 4, b = ) x x 1 = (x + )(x 4). The order of the factors in the answer is unimportant. Check back to Section 1 Exercise 7.1 (i) for the reverse problem (expansion). When factorizing a more complicated quadratic such as 6x + 7x 0, we apply the same general principles, though the problem is harder. We have x + 5 6x + 7x 0 = (x + 5)(x 4) x - 4 Exercises 7.: Factorize (i) x + 4x + (ii) x 7x + 1 (iii) x x 8 (iv) x + x (v) 6x 7x 0 (vi) 1x x 6 (vii) x 5 (viii) x 6x + 9 (ix) x + ax + a (x) 4x 4x + 1 7
4 7. Functional Notation In everyday language, the statement the number of hours of daylight depends on the time of year could also be expressed as the number of hours of daylight is a function of the time of year. In Mathematics, the word function is used with essentially the same meaning, though somewhat idealized. For example, in y = x x + 5, to a particular value of x there corresponds a unique value of y (e.g. x = 1 gives y = 4). We then say that x x + 5 is a function of x and, for convenience, replacing y by f(x), write f(x) = x x + 5. Read the notation f(x) as f of x where f is a symbol for the particular function of x under consideration. Note: f(x) does not mean f x, i.e. the brackets do not signify multiplication. Any symbol other than f can be used for a function. The value of f(x) when x = say is written f(). Example: Find the value of (i) f(x) = x 4 when x = 1 (ii) f(x) = x + 4x 7 when x =. Solutions: ( ) (i) 1 f = 1 4 = = 1 (ii) f() = ( ) + (4 ) 7 = = 7 4
5 Test for a Factor of f(x): If f(a) = 0, then x a is a factor of f(x). Example: Let f(x) = x 7x + 1 Then f() = = = 0 and f(4) = = = 0 x 7x + 1 = (x )(x 4) (see Exercise 7. (ii), Section above) Exercises 7.: (i) Find the value of f(x) = x + 4x 9 when (a) x = (b) x = 0 (c) x = 1 (d) x = 1 (ii) Test whether x is a factor of (a) g(x) = x x (b) h(x) = x + x 10 Note: A cubic expression has an index in the variable, e.g. x + x 0x is a cubic expression (and it happens to factorize, namely, x + x 0x = x(x } + {{ x 0 } ) = x(x + 5)(x 4)). quadratic It is worth noting, as you can readily check, that { x a = (x a)(x + ax + a ) x + a = (x + a)(x ax + a ) 7 5
6 x 1 = (x 1)(x + x + 1) e.g. x + 8 = (x + )(x x + 4) 8x 7 = (x )(4x + 6x + 9). 7.4 Solving a Quadratic Equation Sometimes a quadratic equation has factors in the quadratic expression. In this case it is easy to solve the equation. Example: Solve x x 1 = 0 Solution: Now x x 1 = (x + )(x 4) (See Topic 7, Section ) (x + )(x 4) = 0 i.e. x + = 0 or x 4 = 0 x =, 4 i.e., there are two solutions of a quadratic equation. Checking gives ( ) ( ) 1 = = 0, and = = 0, so the answer is correct. Solving x 4 = 0 = (x )(x + ), we get x =, which is also written x = ±. Or, taking square roots of both sides of x = 4 (i.e. x 4 = 0), we have x = + 4, 4 = ± 4 = ±. Note that = 4 and also ( ) ( ) = 4. We now obtain a useful formula for solving any quadratic equation ax + bx + c = 0. An Important Formula: If ax + bx + c = 0, then x = b ± b 4ac a Proof: Multiply both sides of ax + bx + c = 0 by 4a to get 4a(ax + bx + c) = 4a 0 4a x + 4abx + 4ac = 0 4a x + 4abx = 4ac Add b to both sides to turn the left-hand side into a perfect square i.e. 4a x + 4abx + b = b 4ac 7 6 (ax + b) = b 4ac
7 Take the square root of both sides. We have ax + b = ± b 4ac = b 4ac, b 4ac ax = b ± b 4ac x = b ± b 4ac a Note: that all of b ± b 4ac is divided by a. Example: Solve x 9x + 5 = 0 Solution: In the formula, substitute a =, b = 9, c = 5 to get x = ( 9) ± ( 9) 4 5 = 9 ± = 9 ± 1 6 = 9 + 1, Of course, if we apply the formula to a quadratic expression which factorizes, the calculations will produce a perfect square under the square root sign, and so will disappear from the answer. Thus, in the Example at the beginning of Section 4, in x x 1 = 0 we have a = 1, b = 1, c = 1. The formula yields x = ( 1) ± ( 1) 4 1 ( 1) 1 = 1 ± = 1 ± 49 = 1 ± 7 = 8, 6 = 4, as obtained in the Example. 7 7
8 Exercises 7.4: Solve (i) x 5x + = 0 (ii) 6x 7x = 0 (iii) x x 1 = 0 (iv) t + 5t + 1 = Operations with Square Roots Here we deal briefly with simple arithmetical operations involving square roots. representative examples are chosen. Example: Simplify (a) 8 (b) 6 Solution: Some (a) 8 = = = since = (b) 6 = = i.e. the square root of a product is the product of the square roots. So, generally, ab = a b. Similarly, a b = a b. Example: Simplify 1. Solution: Multiply by (= 1) 1 = 1 = since ( ) = This process is called rationalising the denominator, i.e. the denominator,, which is irrational, is replaced by a denominator () which is a rational number. 7 8
9 Exercises 7.5: Simplify (i) 7 (ii) (iii) 10 (iv) 1 (v) 5 (vi) 14 (vii) 7 1 Exercises 7.6: Solve (i) x 4x = 0 (ii) w 8w + = 0 Topic Revision Exercises 7.7: (i) Expand (x + )( x) (ii) Factorize 10x + 7x 1 (iii) If f(x) = 8x + x find the value of (a) f ( ) 1 (b) f ( 1 ) (iv) Test whether x is a factor of x x 6 (v) Simplify 48 6 (vi) Solve (a) x 9 = 0 (b) x 8x + = 0 (c) 4x 4x + 1 = 0 7 9
10 7.6 Answers to Exercises 7.1: (i) x x 1 (ii) 6x 1x + 6 (iii) x + 7x (iv) xy + x + y + (this is still a quadratic as it involves the product xy of two variables (x, y)) (v) 4x 1x + 9 (vi) x 1 (vii) 49x 16 (viii) x + (a + b)x + ab (ix) x a (x) x ax + a (xi) x + 5xy 1y (xii) 4x 0xt + 5t 7.: (i) (x + 1)(x + ) (ii) (x )(x 4) (iii) (x + )(x 4) (iv) (x 1)(x + ) 7.: (v) (x + 4)(x 5) (vi) (x + )(4x ) (vii) (x + 5)(x 5) (viii) (x ) (ix) (x + a) (x) (x 1) (i) (a) f() = 7 (b) f(0) = 9 (c) f ( 1 ) = 6 1 (d) f( 1) = 11 (ii) (a) g() = 0 so x is a factor, the other factor being x + 1; (b) no, it is not a factor since h() =
11 7.4: (i) x = 1, (ii) x = 1 1, 1 (iii) x = 1+ 5, 1 5 (iv) t = 5+ 17, : (i) (iii) 5 (v) 5 5 (vii) (ii) 4 (iv) (vi) 7 7.6: (i) x = , 1 10 (ii) w = 4+ 7, : (i) 6x 5x + 6 (ii) (x + )(5x 4) (iii) (a) 1 (b) 7 1 (iv) If g(x) = x x 6, then g() = ( 0) so x is not a factor of g(x) (v) (vi) (a) x =, (b) x = 4± 10 (c) x = 1, 1 (a repeated solution as 4x 4x + 1 = (x 1), i.e. a square.) 7 11
1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style
Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with
More informationis identically equal to x 2 +3x +2
Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3
More informationFACTORISATION YEARS. A guide for teachers - Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project
9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers - Years 9 10 June 2011 Factorisation (Number and Algebra : Module
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationTool 1. Greatest Common Factor (GCF)
Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
More information3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
More informationFind all of the real numbers x that satisfy the algebraic equation:
Appendix C: Factoring Algebraic Expressions Factoring algebraic equations is the reverse of expanding algebraic expressions discussed in Appendix B. Factoring algebraic equations can be a great help when
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationis identically equal to x 2 +3x +2
Partial fractions.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as + for any
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 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 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 information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationSOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen
SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen A. GENERALITIES. When a given quadratic equation can be factored, there are 2 best methods
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 informationChapter 7 - Roots, Radicals, and Complex Numbers
Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More informationFactoring Polynomials and Solving Quadratic Equations
Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3
More informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationTim Kerins. Leaving Certificate Honours Maths - Algebra. Tim Kerins. the date
Leaving Certificate Honours Maths - Algebra the date Chapter 1 Algebra This is an important portion of the course. As well as generally accounting for 2 3 questions in examination it is the basis for many
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationSolving Cubic Polynomials
Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial
More informationAlgebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More informationName Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate
More informationALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals
ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an
More informationPartial Fractions Examples
Partial Fractions Examples Partial fractions is the name given to a technique of integration that may be used to integrate any ratio of polynomials. A ratio of polynomials is called a rational function.
More informationFACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1
5.7 Factoring ax 2 bx c (5-49) 305 5.7 FACTORING ax 2 bx c In this section In Section 5.5 you learned to factor certain special polynomials. In this section you will learn to factor general quadratic polynomials.
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.
More informationSchneps, Leila; Colmez, Coralie. Math on Trial : How Numbers Get Used and Abused in the Courtroom. New York, NY, USA: Basic Books, 2013. p i.
New York, NY, USA: Basic Books, 2013. p i. http://site.ebrary.com/lib/mcgill/doc?id=10665296&ppg=2 New York, NY, USA: Basic Books, 2013. p ii. http://site.ebrary.com/lib/mcgill/doc?id=10665296&ppg=3 New
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
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 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 informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More information3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes
Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to
More information0.4 FACTORING POLYNOMIALS
36_.qxd /3/5 :9 AM Page -9 SECTION. Factoring Polynomials -9. FACTORING POLYNOMIALS Use special products and factorization techniques to factor polynomials. Find the domains of radical expressions. Use
More informationALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section
ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationMATH 108 REVIEW TOPIC 10 Quadratic Equations. B. Solving Quadratics by Completing the Square
Math 108 T10-Review Topic 10 Page 1 MATH 108 REVIEW TOPIC 10 Quadratic Equations I. Finding Roots of a Quadratic Equation A. Factoring B. Quadratic Formula C. Taking Roots II. III. Guidelines for Finding
More informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
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 Polynomials
Factoring Polynomials 4-1-2014 The opposite of multiplying polynomials is factoring. Why would you want to factor a polynomial? Let p(x) be a polynomial. p(c) = 0 is equivalent to x c dividing p(x). Recall
More informationFactoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1)
Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if
More informationAlum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
More informationUnit 3: Day 2: Factoring Polynomial Expressions
Unit 3: Day : Factoring Polynomial Expressions Minds On: 0 Action: 45 Consolidate:10 Total =75 min Learning Goals: Extend knowledge of factoring to factor cubic and quartic expressions that can be factored
More information4.1. COMPLEX NUMBERS
4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More information9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.
9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationMESSAGE TO TEACHERS: NOTE TO EDUCATORS:
MESSAGE TO TEACHERS: NOTE TO EDUCATORS: Attached herewith, please find suggested lesson plans for term 1 of MATHEMATICS Grade 11 Please note that these lesson plans are to be used only as a guide and teachers
More informationSOLVING QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD
SOLVING QUADRATIC EQUATIONS BY THE DIAGONAL SUM METHOD A quadratic equation in one variable has as standard form: ax^2 + bx + c = 0. Solving it means finding the values of x that make the equation true.
More informationSIMPLIFYING SQUARE ROOTS
40 (8-8) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More information5. Factoring by the QF method
5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the
More informationPOLYNOMIALS and FACTORING
POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationExamples of Tasks from CCSS Edition Course 3, Unit 5
Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More information1.4. Removing Brackets. Introduction. Prerequisites. Learning Outcomes. Learning Style
Removing Brackets 1. Introduction In order to simplify an expression which contains brackets it is often necessary to rewrite the expression in an equivalent form but without any brackets. This process
More informationFactoring. Factoring Monomials Monomials can often be factored in more than one way.
Factoring Factoring is the reverse of multiplying. When we multiplied monomials or polynomials together, we got a new monomial or a string of monomials that were added (or subtracted) together. For example,
More informationDecomposing Rational Functions into Partial Fractions:
Prof. Keely's Math Online Lessons University of Phoenix Online & Clark College, Vancouver WA Copyright 2003 Sally J. Keely. All Rights Reserved. COLLEGE ALGEBRA Hi! Today's topic is highly structured and
More informationA Systematic Approach to Factoring
A Systematic Approach to Factoring Step 1 Count the number of terms. (Remember****Knowing the number of terms will allow you to eliminate unnecessary tools.) Step 2 Is there a greatest common factor? Tool
More information1 Shapes of Cubic Functions
MA 1165 - Lecture 05 1 1/26/09 1 Shapes of Cubic Functions A cubic function (a.k.a. a third-degree polynomial function) is one that can be written in the form f(x) = ax 3 + bx 2 + cx + d. (1) Quadratic
More informationAlgebra I Credit Recovery
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More 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 informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationSection 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
MODULE - 1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationIndices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková
Indices and Surds The term indices refers to the power to which a number is raised. Thus x is a number with an index of. People prefer the phrase "x to the power of ". Term surds is not often used, instead
More informationApplication. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.
Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:
More informationWelcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move
More informationMathematics Curriculum
Common Core Mathematics Curriculum Table of Contents 1 Polynomial and Quadratic Expressions, Equations, and Functions MODULE 4 Module Overview... 3 Topic A: Quadratic Expressions, Equations, Functions,
More informationAlgebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only
Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials
More informationDifferentiation and Integration
This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have
More informationFACTORING QUADRATIC EQUATIONS
FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods
More informationMathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)
( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =
More informationFactorising quadratics
Factorising quadratics An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to
More information2 : two cube. 5 : five cube. 10 : ten cube.
Math 105 TOPICS IN MATHEMATICS REVIEW OF LECTURES VI Instructor: Line #: 52920 Yasuyuki Kachi 6 Cubes February 2 Mon, 2015 We can similarly define the notion of cubes/cubing Like we did last time, 3 2
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationACCUPLACER Arithmetic & Elementary Algebra Study Guide
ACCUPLACER Arithmetic & Elementary Algebra Study Guide Acknowledgments We would like to thank Aims Community College for allowing us to use their ACCUPLACER Study Guides as well as Aims Community College
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 information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More information