SOLVING QUADRATIC EQUATIONS  COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen


 Philip Peters
 5 years ago
 Views:
Transcription
1 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 to solve it: the ac factoring method (You Tube.com) and the new Diagonal Sum Method (Amazon ebook 2010). This article explains, in general, these two methods, and then compares them through specific examples. 1. The ac method. This method only applies to quadratic equations that can be factored. It is also called the Box Method with a little variation (You Tube). Its concept is to make the given quadratic equation be factored into 2 binomials in x by replacing the term (bx) by 2 terms (b1x) and (b2x) that satisfy the 2 conditions:  1) the product b1.b2 = ac;  2) the sum (b1 + b2) = b. Example 1. Solve: 5x^2 + 6x 8. Solution. Find 2 numbers that the product is (ac = 40) and the sum is b = 6. Proceeding: [(1, 40),(1, 40),(2, 20),(2, 20),(4, 10) OK]. Next, substitute in the equation the term (6x) by the 2 terms (4x) and (10x) and then put in common factor: 5x^2 4x + 10x 8 = 5x(x + 2) 4(x + 2) = (x + 2)(5x 4) = 0 Next, solve the 2 binomials for x: (x + 2) = 0 x = 2 (5x 4) = 0 x = 4/5 2. The Diagonal Sum method. Its concept is direct finding the 2 real roots, in the form of 2 fractions, knowing their sum (b/a) and their product (c/a). It uses 2 rules. The first rule is the Rule of Signs that shows the signs (+ or ) of the 2 real roots before proceeding solving. The second rule is called the Rule for the Diagonal Sum. To know how to use this new Diagonal Sum Method, see the book titled: New methods for solving quadratic equations and inequalities (Amazon ebook 2010) Recall the Rule of Signs.  If a and c have opposite signs, the 2 real roots have opposite signs. Example: the equation 7x^2 5x  12 = 0 has 2 real roots (1) and (12/7) that have opposite signs Page 1 of 7
2  If a and c have the same sign, both real roots have the same sign. If a and b have same sign, both real roots are negative. If a and b have opposite signs, both real roots are positive. Example: the equation (6x^2 + 13x + 7 = 0) has 2 real roots (1) and (7/6) both negative Example: The equation (5x^29x + 4 = 0) has 2 real roots 1 and 4/5, both positive The Rule of The Diagonal Sum Given a pair of 2 real roots (c1, c2). Their product is: c1c2 = c. a1 a2 a1a2 a Their sum is: (c1/a1 + c2/a2) = (c1a2 + c2a1)/a1.a2 = b/a The sum (c1a2 + c2a1) is called the Diagonal Sum of a root pair. The Diagonal Sum of a true real root pair must equal to ( b). If it equals (b), the answers are the negative of the pair. If a is negative, the Rule is reversal in signs. How to proceed with the Diagonal Sum Method. This new method directly selects the probable root pairs from the (c/a) setup by applying in the same time the Rule of Signs. Then, it uses mental math to calculate their diagonal sum and find the one that equals to b (or b). If no root pairs equal (b) or (b), then the equation can t be factored, and the quadratic formula must be used for solving. This new method may be called: The c/a Method. Example 2. Solve: 5x^2 + 6x 8 = 0. Solution. The Rule of Signs indicates roots have opposite signs. The (c/a) setup: (1, 8),(2, 4) (1, 5) If the roots have opposite signs, by convention, always put the negative sign () in front of the first number of the factors of c. The denominator factors are always kept positive. First, you can eliminate the pair (1, 8) because it gives odd number diagonal sums (while b = 6 is even). The remainder (c/a) is: (2, 4)/(1, 5) that gives 2 probable root pairs: (2, 4) and (2, 4) The diagonal of the first pair is: = 6 = b. The 2 real roots are: 2 and 4/5. Example 3. Solve: 5x^2 36x + 7 = 0 Solution. Both real roots are positive. There are 2 probable root pairs: (1, 7) ; (1, 7) The first one can be ignored since 1 is not a real root. The diagonal sum of the second pair is: = 36 = b. The 2 real roots are 1/5 and 7. Example 4. Solve: 8x^2 + 18x + 7 Page 2 of 7
3 Solution. Both roots are negative. The c/a setup: (1, 7)/[(1, 8)(2, 4)]. There are 3 probable root pairs: (1, 7); (1, 7), (1, 7) The second diagonal sum is: = 18 = b. The 2 real roots are 1/2 and 7/4 Note. Before finding probable root pairs, you may first eliminate the pair (1, 8) because it will give an odd number diagonal sum (while b = 18 is even). The c/a setup remainder (1, 7)/(2, 4) gives 2 probable root pairs (1/2, 7/4) and (1/4, 7/2). The first diagonal sum is: 18 = b. The 2 real roots are 1/2 and 7/4. B. SOLVING QUADRTIC EQUATIONS IN DIFFERENT CASES a. CASE 1. When a = 1 Solving the quadratic equation type: x^2 + bx + c = 0. In this case, solving results in finding 2 numbers knowing their sum (b) and their product c. Example 5. Solve: x^2 9x = Solving by the Diagonal Sum Method. Roots have opposite signs. Write factor pairs of c = and, in the same time, apply the Rule of Signs: (1, 102), (2, 51), (3, 34), (6, 17) Stop! This sum is 17 6 = 9 = b. The 2 real roots are 6 and Solving by the ac method. Find 2 numbers that the product is: ac = and the sum is b = 9. Proceed: [(1, 102), (1, 102), (2, 51),(51, 2),(3, 34),(3, 34),(6, 17), (6, 17), OK]. Next, replace the term (9x) by two terms (6x) and (17x): Solve the 2 binomials for x: x^2 9x 102 = x^2 + 6x  17x 102 = 0 x(x  17) + 6(x  17) = 0 (x  17)(x + 6) = 0 (x + 17) = 0 x = 17 (x 6) = 0 x = 6 3. Remark. In this case, solving by the Diagonal Sum method is simpler and doesn t need factoring. The Rule of Signs, that shows the signs of the 2 real roots before proceeding solving, reduces in half the number of permutations (or test cases). In addition, it saves the time used to solve the 2 binomials for x. Example 6. Solve: x^2 + 28x  96 = Solving by diagonal sum method. The Rule of signs indicates both roots are positive. Page 3 of 7
4 Write factorpairs of ac = 96 after applying the Rule of Signs: (1, 96), (2, 48), (3, 32), (4, 24) This sum is: = 28 = b. According to the Diagonal Sum Rule, when a is negative, the 2 real roots are 4 and Solving by the ac method. Find 2 numbers with product ac = 96, and sum = 28. Proceeding: [(1, 96),(1, 96),(2, 48),(2, 48),(3, 32),(3, 32),(4, 24)(4, 24) OK]. Replace (28x) by (4x) and (24x) in the equation: x^2 + 28x 96 = x^2 + 4x + 24x 96 = 0 x(x 4) + 24(x 4) = (x 4)(24 x) = 0 Next, solve the 2 binomials: x 4 = 0 x = 4 24 x = 0 x = 24 b. CASE 2. When a and c are prime numbers. Example 7. Solve: 7x^2 76x 11 = Diagonal Sum Method. Roots have opposite signs. Both a and c are prime. There is unique probable rootpair: (1, 11), since (1) is not a real root. 7 1 Its diagonal sum is 771 = 76 = b. The 2 real roots are 1/7 and The factoring ac method. Find 2 numbers that the product is: ac = 77, and the sum b = 76. Proceeding: [(1, 77),(, 77)]. Next, replace the term (76x) by the 2 terms (1x) and (77x). 7x^2 + x 77x 11 = 0 7x(x 11) +(x 11) = (x 11) (7x + 1) = 0 Solve the 2 binomials for x (x 11) = 0 x = 11 (7x + 1) = 0 x = 1/7 3. Remark. In this case, solving by the Diagonal Sum Method is faster, since there is only one diagonal sum to find. c. CASE 3. When a and c are small numbers and may contain themselves one or 2 factors Example 8. Solve: 8x^2 22x 13 = 0. Page 4 of 7
5 1.The Diagonal Sum method. Roots have opposite signs. Write the (c/a) setup: (1, 13)/[(1, 8),(2, 4)]. Eliminate the pair (1, 8) because it will give oddnumber diagonal sum (while b is even). It remains 2 probable rootpairs: (1, 13) and (1, 13) The first diagonal sum is: = 22 = b. The 2 real roots are: 1/2 and 13/4. 2.The factoring ac method. Find 2 numbers, that their product is ac = 104, and their sum is Proceeding: [(1, 104),(1, 104),(2, 52),(2, 52), (4, 26),(4, 26)]. Replace the term 22x by the 2 terms (4x) and (26x). 8x^2 + 4x 26x 13 = 0. 4x(2x + 1) 13(2x +1) = 0 2x + 1) (4x 13) = 0 Next, solve the 2 binomials: 2x + 1 = 0 x = 1/2 4x 13 = 0. X = 13/4 d. CASE 4. When a and c are large numbers and may contain themselves a few factors These cases are considered complicated because there are many permutations involved. The Diagonal Sum Method may transform a complicated multiple steps solving process into a simplified one by doing a few elimination operations. Example 9. Solve: 12x^2 + 5x 72 = 0. 1.Diagonal Sum method. Roots have opposite signs. In this case, do not directly write down the probable rootpairs because there are too many of them. First, create the (c/a) setup, with all factor pairs of c and of a, and in the same time apply the Rule of Signs: c = (1, 72)(2, 36)(3, 24)(4, 18)(6, 12)(8, 9). a = 12 (1, 12) (2, 6) (3, 4) Before solving, look to eliminate the pairs that do not fit. First, eliminate the pairs: (2, 36), (4, 18),(6, 12) from the numerator and the pair (2, 6) from the denominator because they give even number diagonal sums (while b = 5 is odd number). Next, eliminate the pairs (1, 72) (3, 24)/(1, 12) because they give large diagonal sums (while b= 5). The remainder (c/a) setup is: (8, 9)/ (3, 4). This gives 2 probable rootpairs: (8/3, 9/4) and (8/4, 9/3). The diagonal sum of the first pair is: = 5 = b. The 2 real roots are: 8/3 and 9/4. Page 5 of 7
6 2..The factoring ac method. In these cases, solving becomes inconvenient because the product ac is a large number. Find 2 numbers: Product ac = Sum: b = 5. Proceeding: [(1, 864),(1, 864),(2, 432),(2, 432),(3, 288), (18, 48),(18, 48),(24, 36),(24, 36),( 32, 27),(32, 27) OK]. Next, replace the term 5x by the 2 terms 27x and 32x. 12x^2 27x + 32x 72 = 0. 3x(4x 9) + 8(4x  9) = 0 (4x 9)(3x + 8) = 0 Next, solve the 2 binomials: 4x 9 = 0 x = 9/4. 3x + 8 = 0 x = 8/3. Example 10. Solve: 24x^2 + 59x + 36 = Diagonal Sum method. Both roots are negative. The (c/a) setup: (1, 36)(2, 18)(3, 12)(4, 9)(6, 6) (1, 24)(2, 12) (3, 8) (4, 6) Before proceeding, eliminate the pairs (2, 18),(6, 6)/(2, 12),(4, 6) because they give evennumber diagonal sums (while b = 59 is odd). Also, eliminate the pairs: (1, 36)(3, 12)/(1, 24) because they give large diagonal sums, as compared to b = 59. The remainder (c/a) is: (4, 9) (3, 8) This gives 2 probable rootpairs: (4, 9) and (4, 9). The diagonal sum of the first pair is: = 59 = b. The 2 real roots are 4/3 and 9/8. 2.The ac method. Find 2 numbers: product ac = 864; sum b = 59. Proceeding [(1, 864)(1, 864),(2, 432),(2, 432),(4, 216),(4, 216), (18,  48)(18, 48)(24, 36) (24, 36),(27,  32)(27, 32)]. Since the product ac = 864 is too large, the proceeding takes too much time to complete. Next, replace term (59x) by the 2 term( 27x) and (32x). 24x^2 + 27x + 32x + 36 = 0. 8x(3x + 4) + 9(3x + 4) = 0 (3x + 4)(8x + 9) = 0 3x + 4 = 0  x = 4/3 8x + 9 = 0  x = 9/8. Page 6 of 7
7 A. CONCLUSION AND REMARKS. 1. Both methods deserve to be studied because they provide students with opportunities to improve math skills and logical thinking that are the ultimate goals of learning math. 2. When the constants a and c are large numbers and may contain themselves many factors, then students are advised to use the quadratic formula for solving. However, either performing mentally or by calculator, remember to always proceed solving in 2 steps. First step, compute the Discriminant D = b^2 4ac. If the given quadratic equation can be factored, then D must be a perfect square. Second step, compute algebraically the rest of the formula with the value of D s square root that should be a whole number. Make sure that the 2 real roots be in the form of 2 fractions and not in decimals. If calculators are not allowed, solving by the Diagonal Sum Method may be faster and better. 3. Best methods to solve quadratic equations. The quadratic formula is obviously the best choice to solve a quadratic equation in standard form ax^2 + bx + c = 0, especially when calculators are allowed. However the ultimate goal of math learning isn t only to solve equations with calculators. The math learning process wants students to learn solving quadratic equations by a few other methods in order to improve their math skills and their logical thinking. Although 99% (?) of the quadratic equations in real life are not factorable, many math equations in books/tests/exams are intentionally setup so that students must solve them by the factoring method. So far, the 2 best methods to solve factorable quadratic equations are the ac method and the new Diagonal Sum Method. (This article was written by Nghi H Nguyen, the coauthor of the new Diagonal Sum Method for solving quadratic equations) Page 7 of 7
SOLVING 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 informationSOLVING QUADRATIC EQUATIONS  COMPARE THE FACTORING AC METHOD AND THE NEW TRANSFORMING METHOD (By Nghi H. Nguyen  Jan 18, 2015)
SOLVING QUADRATIC EQUATIONS  COMPARE THE FACTORING AC METHOD AND THE NEW TRANSFORMING METHOD (By Nghi H. Nguyen  Jan 18, 2015) GENERALITIES. When a given quadratic equation can be factored, there are
More informationSOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014))
SOLVING QUADRATIC EQUATIONS BY THE NEW TRANSFORMING METHOD (By Nghi H Nguyen Updated Oct 28, 2014)) There are so far 8 most common methods to solve quadratic equations in standard form ax² + bx + c = 0.
More informationBEST METHODS FOR SOLVING QUADRATIC INEQUALITIES.
BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES. I. GENERALITIES There are 3 common methods to solve quadratic inequalities. Therefore, students sometimes are confused to select the fastest and the best
More informationCM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of
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 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 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 informationSHORTCUT IN SOLVING LINEAR EQUATIONS (Basic Step to Improve math skills of high school students)
SHORTCUT IN SOLVING LINEAR EQUATIONS (Basic Step to Improve math skills of high school students) (by Nghi H. Nguyen) Most of the immigrant students who first began learning Algebra I in US high schools
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 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 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 informationCONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form <==> Intercept Form <==> Vertex Form) (By Nghi H Nguyen Dec 08, 2014)
CONVERT QUADRATIC FUNCTIONS FROM ONE FORM TO ANOTHER (Standard Form Intercept Form Vertex Form) (By Nghi H Nguyen Dec 08, 2014) 1. THE QUADRATIC FUNCTION IN INTERCEPT FORM The graph of the quadratic
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 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 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 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 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 informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
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 informationFactoring Polynomials
Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring
More informationEAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors.
EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an
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 informationMATH 90 CHAPTER 6 Name:.
MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a
More informationPartial Fractions. (x 1)(x 2 + 1)
Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +
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 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 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 informationFactoring. Factoring Polynomial Equations. Special Factoring Patterns. Factoring. Special Factoring Patterns. Special Factoring Patterns
Factoring Factoring Polynomial Equations Ms. Laster Earlier, you learned to factor several types of quadratic expressions: General trinomial  2x 25x12 = (2x + 3)(x  4) Perfect Square Trinomial  x
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 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 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 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 informationFactoring Polynomials
Factoring Polynomials 412014 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 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 informationFACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1
5.7 Factoring ax 2 bx c (549) 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 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 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 informationYear 9 set 1 Mathematics notes, to accompany the 9H book.
Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H
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 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 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 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 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 informationIn algebra, factor by rewriting a polynomial as a product of lowerdegree polynomials
Algebra 2 Notes SOL AII.1 Factoring Polynomials Mrs. Grieser Name: Date: Block: Factoring Review Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 3 In algebra, factor by rewriting
More informationFactoring ax 2 + bx + c  Teacher Notes
Southern Nevada Regional Professi onal D evel opment Program VOLUME 1, ISSUE 8 MAY 009 A N ewsletter from the Sec ondary Mathematic s Team Factoring ax + bx + c  Teacher Notes Here we look at sample teacher
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 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 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 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 informationFactors and Products
CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
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 informationAcademic Success Centre
250) 9606367 Factoring Polynomials Sometimes when we try to solve or simplify an equation or expression involving polynomials the way that it looks can hinder our progress in finding a solution. Factorization
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationMATH 108 REVIEW TOPIC 10 Quadratic Equations. B. Solving Quadratics by Completing the Square
Math 108 T10Review 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 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 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 informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
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 informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationTHE TRANSPOSING METHOD IN SOLVING LINEAR EQUATIONS (Basic Step to improve math skills of high school students) (by Nghi H. Nguyen Jan 06, 2015)
THE TRANSPOSING METHOD IN SOLVING LINEAR EQUATIONS (Basic Step to improve math skills of high school students) (by Nghi H. Nguyen Jan 06, 2015) Most of the immigrant students who first began learning Algebra
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 informationUsing the ac Method to Factor
4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trialanderror
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationACCUPLACER. Testing & Study Guide. Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011
ACCUPLACER Testing & Study Guide Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011 Thank you to Johnston Community College staff for giving permission to revise
More information6.4 Special Factoring Rules
6.4 Special Factoring Rules OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor a sum of cubes. By reversing the rules for multiplication
More informationFactoring (pp. 1 of 4)
Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationFactoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai
Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for
More informationALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section
ALGEBRA 2 CRA 2 REVIEW  Chapters 16 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 53.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 64.2 Solving Equations by
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 informationFactoring Special Polynomials
6.6 Factoring Special Polynomials 6.6 OBJECTIVES 1. Factor the difference of two squares 2. Factor the sum or difference of two cubes In this section, we will look at several special polynomials. These
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 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 informationA. Factoring out the Greatest Common Factor.
DETAILED SOLUTIONS AND CONCEPTS  FACTORING POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!
More informationMath 25 Activity 6: Factoring Advanced
Instructor! Math 25 Activity 6: Factoring Advanced Last week we looked at greatest common factors and the basics of factoring out the GCF. In this second activity, we will discuss factoring more difficult
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
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 informationIOWA EndofCourse Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.
IOWA EndofCourse Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of $2,000. She also earns $500 for
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 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 information1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).
.7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational
More informationAlgebra 2: Q1 & Q2 Review
Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short
More informationAIP Factoring Practice/Help
The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There
More informationBy reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.
SECTION 5.4 Special Factoring Techniques 317 5.4 Special Factoring Techniques OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor
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 information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
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 informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
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 informationAlgebra 1. Curriculum Map
Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring
More informationFactoring Flow Chart
Factoring Flow Chart greatest common factor? YES NO factor out GCF leaving GCF(quotient) how many terms? 4+ factor by grouping 2 3 difference of squares? perfect square trinomial? YES YES NO NO a 2 b
More informationSECTION 16 Quadratic Equations and Applications
58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be
More informationFactoring Polynomials
Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,
More information