# A Systematic Approach to Factoring

Size: px
Start display at page: Transcription

1 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 1 (Page 3) (Remember****This is the most important tool. It is often used to hide other tools. When you factor the greatest common factor, it allows you to recognize other tools.) Step 3 Can you group the expression? (Grouping) Tool 2 (page 6) (Remember****Grouping requires 4 terms. If there are four terms present, the object is to group the first two and factor them. You will then have to factor the second two terms and try to make them look the first two terms.) Step 4 Can you use the AC method? Tool 3 (page 7) (Remember****The A times C method requires the problem to have three terms. Then number that we call A when multiplied by the number called C must have a sum that equals the middle term. We call the middle term B. We say that A times C must equal B.) Ax 2 + Bx+ C 1

2 Step 5 Can the problem be factored as the difference of two perfect squares. Tool 4 (Remember****This tool requires two terms. Both the first and second terms must be perfect squares. A minus sign must separate the two terms. A plus + sign cannot separate the difference of two squares.) Step 6 Is the problem the sum/difference of perfect cubes? Tool 5 (Remember****Does the first and second terms have a cube root? This tool requires that a polynomial to have exactly two terms. The problem can either have the sum or the difference of two perfect cubes. Variables that are perfect cubes have exponents that are divisible by three.) 2

3 Polynomial Factoring Factoring polynomials is one of the most important topics covered in Elementary Algebra. We will divide the concept into five tools. Tool 1 Factoring the greatest common monomial: (The most important tool!!!!) A monomial is a one termed polynomial. For example: 3x, 5x 2 t 5, 6, -3v The term factor in this context means to divide. We are going to divide the largest factor found in each term of a polynomial from that polynomial. This is the most important of the five tools that we will discuss. Important Terms: Factor The original definition of the factor is an algebraic expression that is multiplied. Examples: 12 has factors 3 x 4 and 2 x 6 3x 2 y 4 z has factors 3, x, 2 y, 4 z When factors are variables, they are simply algebraic factors. We say in words that this expression is 3 times x 2 times y 4 times z. The key is that these expressions (algebraic expressions) are separated by multiplication. ***Remember, factors are multiplied and factoring is dividing.*** Factoring The changing of an addition/subtraction problem into a multiplication problem. Term A term is an algebraic expression separated by either a plus sign or a minus sign. Example: 3x 2 + 5y 3 z 8m + 7 Constant A number that does not have a variable associated with it. Combine To add or subtract. (Which ever is implied in the problem.) Numerical Coefficient (coefficient) The number that is multiplied by the variable. 3

4 3x 2 + 5y 3 z 8m + 7 This algebraic expression has four terms. The first term 3x 2 has two factors. This implies that a factor is not a terms but factors make up a term. The second term in the expression is 5y 3 z. This term has three factors. The factors are 5, y, 3 and z. These factors are separated by an understood multiplication symbol. ***Remember, factors are multiplied.*** Factoring the greatest common monomial 3x 2 yz 3 6x 3 y 4 z xy 2 z 5 The term common implies that eh factors that you are looking for have something in common. You must first understand that there are three terms in this expression. 3x 2 yz 3 is the first term. 6x 3 y 4 z 2 is the second term, and 12xy 2 z 5 is the third term. You are looking for the largest factor that is common to all three terms. In the end, you will divide each of these terms by this greatest common factor, (greatest common monomial) that you are trying to create. You are looking for common factors in each term. 3x 2 yz 3 m 6x 3 y 4 z xy 2 z 5 m 3x 2 yz 3 m 6x 3 y 4 z xy 2 z 5 m 3x 2 yz 3 m 6x 3 y 4 z xy 2 z 5 m 3x 2 yz 3 m 6x 3 y 4 z xy 2 z 5 m The numerical coefficients 3, 6, and 12 have a greatest common factor of 3. This 3 becomes a part of the greatest common factor. The variable x 2, x 3, and x are common variables in that they are all x s. The smallest variable {x} is the only one that can divide evenly into each term. This means that {x} is part of the greatest common factor. The variables y, y 4, and y 2 are all common in that each term has at least one variable {y}. The smallest number of y s present is y 1 or simply y. this y becomes part of the greatest common factor. The variables z 3, z 2, and z 5 are all common in that each term has at least one variable {z}. The smallest number of z s present is z 2. This variable z 2 becomes the final part of the greatest common factor. The last factor in the first term is m. The variable m cannot be a part of the greatest common factor because there is not at least one m in each of the terms. 4

5 The greatest common factor (GCF) is 3xyz 2. You will divide each term by this GCF. ***Remember when you divide polynomials, you actually divide the numerical coefficients, but you simply subtract the exponents of the variables.*** The result is xzm from the first term, 2x 2 y 3 from the second term, and 4yz 3 m from the third term. The answer should be written as follows: (The signs stay consistent.) 3xyz 2 ( xzm - 2x 2 y 3 + 4yz 3 m ). You should recognize that this problem has been factored because it is a multiplication problem. (****Remember, Factoring is the changing of an addition/subtraction problem into a multiplication problem. This rule carries over into any type of polynomial. When you are looking for a GCF this is what should be done. Again, it must be said that this is the most important of the five tools that we are going to discuss. It can be a part of each of the remaining four tools. Simplify the following. 5x 2 (3m + 1) 7x 5 (3m + 1) + 4x 3 (3m + 1) 5x 2 (3m + 1) 7x 5 (3m + 1) + 4x 3 (3m + 1) This polynomial has three terms. The minus sign separates the first two terms ad the plus sign separates the last two terms. The first term has three factors. The first factor is 5, the second is x 2, and the third factor is the expression (3m + 1). You should notice right away that the factors 5, 7, and 4, are all numerical coefficients, but they do not share a GCF. This implies that my overall GCF will not have a numerical coefficient. Each term has {x} to some degree. The first term has x 2 which implies that there are two x s. The second term has five x s and the third term has three x s. When we ask, What is the largest number of x s present in each term? You can see right away that the number in question is not going to be the largest number of x s present overall. The largest number of x s present is x 5, but you can only divide each term by x 2 because you cannot subtract 5 from 2 and have a positive exponent. Obviously, the expression (3m + 1) is common in all three terms. ****Remember, the expression (3m + 1) is a factor just the same as any other factor; you must treat it as you would any other factor.**** Solution: x 2 (3m + 1) (5-7 x 3 + 4x) 5

6 Tool 2 Grouping Grouping Grouping requires four or more terms. However, there must be an even number of terms. When using Tool 2, you must guarantee that you cannot use Tool 1 first. Procedure: 6x 2 4x 3xa +2a Group the first two terms. (6x 2 4x) Then group the second two terms. ( 3xa +2a) Find the GCF for the first two terms. Then find the GCF for the second two terms. 2x(3x 2) - a(3x 2) In this case you must factor {-a}. Only factoring {-a} will make the two expressions have something in common. Notice, factoring {-a} produces expressions which look exactly the same. You are left with a two termed expression: 2x(3x 2) - a(3x 2) You can now factor the expression (3x 2) from both terms. That completes Tool 2. (3x 2) (2x a) 6

7 Tool 3 The AC Method Tool 3 is called the AC method. This is short for the A * C method. The AC method requires that a polynomial have exactly three terms. When we use Tool 3, we are simply taking a three termed polynomial and changing it into a four termed polynomial so we can group factor it by grouping (Tool 2). We will introduce the quadratic equation in standard form: Ax 2 + Bx + C 6x 2 - x - 1 You will notice that the number that represents {A} is 6 and the number that represents {C} is -1. So A = 6 and C = -1 When we say A * C, what we are really saying is 6 * -1. In this case A * C = -6. Now we must find factors of {-6}: Factors: 1 * 6 2 * 3 You must look for the factors of {-6} whose sum produces the coefficient of the middle term. The middle term s coefficient is {-1}. The only way to get {-1} is by adding the factors in a very specific way. The factors {1, 6} cannot be combined (added/subtracted) to equal -1. However, the factors {2, 3} can be added to result in a -1 only when the combination is present. So only when 3 is negative and 2 is positive will we obtain 1 as the difference. The rule says that the largest factor must have the same sign as the middle term. You must then replace the x (the middle term) with its equivalent expression. - x = - 3x + 2x 6x 2 - x - 1 6x 2-3x + 2x - 1 You will notice that you have a nice three termed polynomial into a four termed polynomial. This changed was made so that we can now use the grouping method (Tool 2). (6x 2-3x) + (2x 1) 3x(2x - 1) +1(2x 1) 7

8 (3x + 1) (2x 1) Factored! ****Remember**** Grouping requires that the problem to have four terms and an {A * C} has three terms. Logic says that if you want to group, you must make this three termed polynomial into a four termed polynomial. Replacing the middle term with two terms accomplishes this requirement. Examples: Algebraic Expression A * C Replacement Factors for Middle Term Results 12x 2-17x * 5-12x 5x 12x 2-12x 5x + 5 6x xy + 7y 2 6 * 7 42xy - xy 6x xy - xy + 7y 2 2x 2-11x * 40-16x + 5x 2x 2-16x + 5x y 2 15y * 9-12y - 3y 4y 2-12y - 3y + 9 Tool 4 The Difference of Two Perfect Squares The difference of two perfect squares is Tool 4. Understanding this tool requires an understanding of the words perfect square and square root. Perfect Square A perfect square is a number that is created by multiplying a number times itself. Example: * 2 = 4 4 is a perfect square because it is created by multiplying the number two times itself. 3 * 3 = 9 9 is a perfect square because it is created by multiplying the number three times itself. 8

9 Perfect square numbers are created by multiplying a number times itself. Perfect square variables simply have exponents that are even. Square Root A square root is the number that is multiplied times itself to produce a perfect square. Number is a square root because Multiply Perfect Square 7 is a square root because 7 * is a square root because 5 * 5 25 x is a square root because x * x x 2 x 3 is a square root because x 3 * x 3 x 6 ***Remember*** To find the square root of a variable, simply divide the exponent by two. This implies that the exponents must be even. Only even numbers are divisible by two. Examples: x 6 25x 2 y 4 9z 4 x 4 Number is a square root because Multiply 7 is a square root because 7 * 7 5 is a square root because 5 * 5 x is a square root because x * x x 3 is a square root because x 3 * x 3 9

10 When we find the difference of two perfect squares, we must first recognize a perfect square. Recognition is the key!!! Example: 9x 2 y 2 25 z 2 36x 4 z (x + 7y) 2-49(x + 7y) 2 All problems that qualify to be called the difference of two perfect squares, (difference of squares, for short) must have only two terms. These terms must be separated by a minus sign. The sum of squares cannot be factored. When we see that the problem is difference of two perfect squares, we must treat them all exactly the same way. We make two sets of parenthesis: Step 1 ( ) ( ) We put a plus sign in one and a Step 2 ( + ) ( - ) minus sign in the other set of parenthesis. The order does not matter. Example: 25x 2 y 4 We find the square root of the Step 3 ( 5x + ) ( 5x - ) first term and write it as the first term in both parenthesis. We then write the square root of Step 4 ( 5x + y 4 ) ( 5x - y 4 ) The second term as the second term in both parenthesis. Examples: 25y 2 36 = ( 5y + 6 ) ( 5y - 6 ) 16y 2 z 2 49 = ( 4yz + 7 ) ( 4yz - 7 ) Tool 4 The Sum & Difference of Two Perfect Cubes Perfect cube A perfect cube is a number that is obtained by multiplying a number times itself three times. 10

11 Example: Cube root: A cube root is the number that is multiplied times itself three times itself to create a perfect cube. Examples: The numbers: 2 is the cube root of 8. 3 is the cube root of * 3 * 3 = 27 so the cube root of 27 is 3. 4 * 4 * 4 = 64 so the cube root of 64 is 4. Like the exponents of the variables of perfect squares must be divisible by two, the exponents of the variables of perfect cubes must be divisible by three. This implies that the following variables are perfect cubes. Examples: x 3 x 15 y 3 z 9 The following terms are perfect cubes: Examples: 27x 3 8y 3 v 9 64x 3 m 6 ***Remember*** The numerical coefficients (the number that is multiplied by the variable), must meet a different requirement than a variable to be called a perfect cube. The numerical coefficient must have a cube root and all variables must have exponents that are divisible by three. When we factor the difference of two perfect cubes, we must make two sets of parenthesis. They must be of different sizes because a two termed polynomial (binomial) will go in the first set and the second set will house a three termed polynomial (trinomial). Example: 8x 3-27 Because of the minus sign and the fact that both terms are perfect cube, this is going to be the difference of two perfect cubes. Recognition is the key!!! Step 1 Two sets of parenthesis ( ) ( ) The parenthesis must not be of equal size. One must house a binomial a two termed polynomial, and the other must house trinomial a three termed polynomial. Step 2 Because this is the difference of two perfect cubes, the signs that must go in the parenthesis must be very specific. The terms of the binomial are separated by a minus sign and the terms of the trinomial are separated by two plus signs. ( - ) ( + + ) 11

12 Step 3 8x 3-27 = ( 2x - 3 ) ( + + ) Cube root of the first term^ ^Cube root of the second term The rule says that you must write the cube root of the first as the first and the cube root of the second as the second. Since the cube root of the first term is 2x, is must be written as the first term of the binomial. The cube root of the second term, which is 3, must be written as the second term in the binomial. To fill the three spaces of the trinomial, you will not use the original problem. You must ignore the expression 8x Instead, you must take the first term 2x and square it. Squaring it means squaring each factor in the term. 2x then becomes 4x 2. 4x 2 becomes the first term in the trinomial. Step 4 8x 3-27 = ( 2x - 3 ) ( 4x ) First term^ ^First term squared To get the last term in the trinomial, you must square the last term in the binomial. Since the last term in the binomial is 3, when you square it, it becomes 9. This implies that 9 becomes the last term in the trinomial. 8x 3-27 = ( 2x - 3 ) ( 4x ) Second term^ ^Second term squared Finally, to get the middle term of the trinomial, you must ignore the signs in the middle two terms in the binomial and just multiply them together. 2x times 3 is 6x. Thus, 8x 3-27 = ( 2x - 3 ) ( 4x 2 + 6x + 9 ) ^The product of the first & last terms in the binomial ***Remember*** Since the signs were already set at the beginning of the problem, you will ignore the signs of the terms in the binomial and just multiply the terms. We say that we just take their absolute values and then multiply them. Formulas: Difference of Two Perfect Cubes ( F 3 L 3 ) = (F 2 + FL + L 2 ) Sum of Two Perfect Cubes ( F 3 + L 3 ) = (F 2 - FL + L 2 ) 12

13 13

### expression 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

### 1.3 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.

### Name 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

### NSM100 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

### Chapter 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

### ( ) 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

### 1.3 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,

### A. 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!

### Greatest Common Factor (GCF) Factoring Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication

### Factoring 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

### Tool 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

### In algebra, factor by rewriting a polynomial as a product of lower-degree 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

### Factoring 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

### Factoring 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

### Alum 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

### Factoring 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

### Section 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

### Operations with Algebraic Expressions: Multiplication of Polynomials Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the

### Factoring Algebra- Chapter 8B Assignment Sheet Name: Factoring Algebra- Chapter 8B Assignment Sheet Date Section Learning Targets Assignment Tues 2/17 Find the prime factorization of an integer Find the greatest common factor (GCF) for a set of monomials.

### Factoring 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

### AIP 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

### FACTORING 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.

### Factoring 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

### Factoring 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

### 5 means to write it as a product something times something instead of a sum something plus something plus something. Intermediate algebra Class notes Factoring Introduction (section 6.1) Recall we factor 10 as 5. Factoring something means to think of it as a product! Factors versus terms: terms: things we are adding

### This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This is Factoring and Solving by Factoring, chapter 6 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

### When factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF. Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property

### Definitions 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 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

### Algebra 1 Chapter 08 review Name: Class: Date: ID: A Algebra 1 Chapter 08 review Multiple Choice Identify the choice that best completes the statement or answers the question. Simplify the difference. 1. (4w 2 4w 8) (2w 2 + 3w 6)

### POLYNOMIALS 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 250) 960-6367 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

### Copy 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,...}

### Math 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

### Finding Solutions of Polynomial Equations DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL EQUATIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### MATH 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

### By 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

### Factoring 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

### Factors 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

### 6.1 The Greatest Common Factor; Factoring by Grouping 386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

### Sect 6.7 - Solving Equations Using the Zero Product Rule Sect 6.7 - Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred

### Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

### FOIL FACTORING. Factoring is merely undoing the FOIL method. Let s look at an example: Take the polynomial x²+4x+4. FOIL FACTORING Factoring is merely undoing the FOIL method. Let s look at an example: Take the polynomial x²+4x+4. First we take the 3 rd term (in this case 4) and find the factors of it. 4=1x4 4=2x2 Now

### Algebra 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

### Factoring. 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,

### Lagrange 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

### Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

### FACTORING POLYNOMIALS 296 (5-40) 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

### FACTORING OUT COMMON FACTORS 278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the

### Factoring. 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 2-5x-12 = (2x + 3)(x - 4) Perfect Square Trinomial - x

### SECTION 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

### Algebra 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

### 6.1 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

### FACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c Tallahassee Community College 55 FACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c This kind of trinomial differs from the previous kind we have factored because the coefficient of x is no longer "1".

### Determinants 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

### Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

### Factoring 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,

### 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. 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

### Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Unit Title: Quadratic Expressions & Equations Course: Algebra I Unit 8 - Quadratic Expressions & Equations Brief Summary of Unit: At

### 6.4 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

### Mathematics Placement Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### Factoring a Difference of Two Squares. Factoring a Difference of Two Squares 284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this

### SPECIAL PRODUCTS AND FACTORS CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

### Factoring - Grouping 6.2 Factoring - Grouping Objective: Factor polynomials with four terms using grouping. The first thing we will always do when factoring is try to factor out a GCF. This GCF is often a monomial like in

### Veterans Upward Bound Algebra I Concepts - Honors Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER

### Algebra 2 PreAP. Name Period Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing

### This 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

### Factoring (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

### Polynomials and Factoring 7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of

### Sect 6.1 - Greatest Common Factor and Factoring by Grouping Sect 6.1 - Greatest Common Factor and Factoring by Grouping Our goal in this chapter is to solve non-linear equations by breaking them down into a series of linear equations that we can solve. To do this,

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

### 15.1 Factoring Polynomials LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE

### Factoring 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

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

### GCF/ Factor by Grouping (Student notes) GCF/ Factor by Grouping (Student notes) Factoring is to write an expression as a product of factors. For example, we can write 10 as (5)(2), where 5 and 2 are called factors of 10. We can also do this

### Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions: Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?

### Math 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

### This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the

### 4.4 Factoring ax 2 + bx + c 4.4 Factoring ax 2 + bx + c From the last section, we now know a trinomial should factor as two binomials. With this in mind, we need to look at how to factor a trinomial when the leading coefficient is

### MATH 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.

### Mathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal

### FACTORING 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.

### Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Algebra 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

### How To Factor By Gcf In Algebra 1.5 7-2 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Simplify. 1. 2(w + 1) 2. 3x(x 2 4) 2w + 2 3x 3 12x Find the GCF of each pair of monomials. 3. 4h 2 and 6h 2h 4. 13p and 26p

### SIMPLIFYING ALGEBRAIC FRACTIONS Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is

### Factoring Trinomials using Algebra Tiles Student Activity Factoring Trinomials using Algebra Tiles Student Activity Materials: Algebra Tiles (student set) Worksheet: Factoring Trinomials using Algebra Tiles Algebra Tiles: Each algebra tile kits should contain

### COLLEGE 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 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

### SOLVING 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

### Polynomial Operations and Factoring Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

### SOLVING 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

### 6.6 Factoring Strategy 456 CHAPTER 6. FACTORING 6.6 Factoring Strategy When you are concentrating on factoring problems of a single type, after doing a few you tend to get into a rhythm, and the remainder of the exercises, because

### Big Bend Community College. Beginning Algebra MPC 095. Lab Notebook Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond

### Gouvernement du Québec Ministère de l Éducation, 2004 04-00813 ISBN 2-550-43545-1 Gouvernement du Québec Ministère de l Éducation, 004 04-00813 ISBN -550-43545-1 Legal deposit Bibliothèque nationale du Québec, 004 1. INTRODUCTION This Definition of the Domain for Summative Evaluation

### MATD 0390 - Intermediate Algebra Review for Pretest MATD 090 - Intermediate Algebra Review for Pretest. Evaluate: a) - b) - c) (-) d) 0. Evaluate: [ - ( - )]. Evaluate: - -(-7) + (-8). Evaluate: - - + [6 - ( - 9)]. Simplify: [x - (x - )] 6. Solve: -(x + Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials 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