# Monomial Factors. Sometimes the expression to be factored is simple enough to be able to use straightforward inspection.

Size: px
Start display at page:

Download "Monomial Factors. Sometimes the expression to be factored is simple enough to be able to use straightforward inspection."

Transcription

1 The Mathematics 11 Competency Test Monomial Factors The first stage of factoring an algebraic expression involves the identification of any factors which are monomials. We will describe the process by working through several examples in detail. Sometimes the expression to be factored is simple enough to be able to use straightforward inspection. Example 1: Factor the expression: 2xy + 6yz. We see that 2xy = 2 x y 6yz = 2 3 y z We can quickly recognize that both terms contain the factors 2 y in common. Thus 2xy + 6yz = 2y(x + 3z). The right h side here is in factored form because it is a single term only. As it is written, it does not consist of two or more parts which are connected by plus or minus signs. It is just a single term which is the product of three factors in this case. That the factored expression above is mathematically equivalent to the original expression is easily verified by multiplication. 2y(x + 3z) = (2y)(x) + (2y)(3z) = 2yx + 6yz = 2xy + 6yz. In fact, you can always verify that a factorization is correct by re-multiplication to confirm that the original expression is regenerated. Example 2: Remove all common monomial factors from: 16x 3 24x x. This expression is a bit more complicated than the one in the previous example. With it, we can begin to implement the systematic method that should be used in all but the very simplest cases where the factorization is immediately obvious. (In fact, you should probably always do the systematic method, because that is really the only way to make sure that what you thought was an easy-to-see simple factorization really is complete correct!) We start by selecting any term in the expression -- it doesn t matter which one we pick, so we will pick the first one here: 16x 3. Enumerate the factors which make up this term: David W. Sabo (2003) Monomial Factors Page 1 of 7

2 16x 3 = 2 4 x 3 This means that any monomial factor common to all three terms in the original expression must either be a power of 2 or a power of x. This is because monomial factors common to all three terms must be factors of each individual term, hence they must be a factor of the first term (or whichever one you picked). So, all we have to do is check what power of 2 is common to all three terms, what power of x is common to all three terms. It s probably easiest at this stage to write out the factorizations of each term for easy reference: 16x 3 = 2 4 x 3 24x 2 = x 2 84x = x 1 (The method for resolving whole numbers into products of prime factors has been described in an earlier document in these notes.) From this we see that all three terms contain a factor of 2 2, so this is a factor of the entire expression all three terms contain a factor of x 1 or x, so this is a factor of the entire expression the above two bullets in this list cover all possible factors of the first term, so cover all possible monomial factors of the entire expression. Thus, 2 2 x = 4x is the most extensive monomial factor common to all three terms of the original expression. We can write 16x 3 24x x = 4x(4x 2 6x + 21) The trinomial in brackets is obtained by removing the factor 4x from each of the three terms in the original expression: 16x 3 4x = 4x 2-24x 2 4x = -6x 84x 4x = 21 The expression on the right above is the required factored expression here. The terms of the trinomial in the brackets contain no further common monomial factors, so all of the common monomial factors have been factored-out as required. You should take a minute here to verify that if you multiply to remove the brackets from the factored expression, you get precisely the original expression back again. Example 3: Remove all common monomial factors from 42b 2 y 28by 2. This example is very similar to the previous one, so you should try it as a practice problem on your own first, before looking at our brief outline of a solution. First, write out the factorization of each of the two terms explicitly: David W. Sabo (2003) Monomial Factors Page 2 of 7

3 42b 2 y = b 2 y 1 28by 2 = b 1 y 2 Notice that we ve written exponents of symbols explicitly, even if those exponents are 1 (just as a visual cue when we check now for common factors between the two terms). It is also helpful to sort the factors of the individual terms in a common order. Here numerical prime factors are sorted from smallest to largest (going left to right) symbolic factors are sorted alphabetically. Comparison of these two factorizations indicates immediately that the common factors are 2, 7, b, y, all to the first power. Thus 42b 2 y 28by 2 = 2 7 b y(3b 2y) = 14by(3b 2y). The terms in the expression in brackets on the right here are obtained by taking what s left of each of the original terms when the common factors are removed. Thus, since 42b 2 y = b 2 y 1 when we remove the factors 2, 7, b 1, y 1, all that s left is the 3 one of the factors b, or 3b. A similar inspection indicates that after removal of these four common factors from 28by 2, all that is left is the factors 2 y, each to the first power. Thus, the required factorization here is 42b 2 y 28by 2 =14by(3b 2y). We ll leave it up to you to verify that this is correct by multiplication. Example 4: Identify all of the common monomial factors in 25x 4 y 3 zw x 3 y 3 zr x 2 y 3 zw This looks bad (somewhat intentional, to demonstrate it is as easily hled using a systematic approach as any of the previous examples). The goal right here is to rewrite this trinomial in the form: (a monomial) (whatever else) Note that this pattern is a single term which is the product of two parts. The a monomial part will be identified here, in the process determining what the whatever else part is. (Before leaving the topic of factoring, we will also demonstrate things to check for factoring the whatever else part a bit further in some cases.) In solving this example, we ll use a slight variation on the methods demonstrated for the previous two examples. We start by selecting any of the three terms to guide the process it doesn t matter which term you choose, though there may be a slight savings in work if a simpler-looking term is used. So, we decide to base this analysis on the parts of the first term, the 25x 4 y 3 zw 2. Next, identify all of the simple factors of the selected term, including prime factors in the numerical coefficient. The term we have chosen to work from has simple factors 5, x, y, z w. David W. Sabo (2003) Monomial Factors Page 3 of 7

4 (Note that if we had chosen to key our analysis to the third term, the 30x 2 y 3 zw, our list of simple factors would be 2, 3, 5, x, y, z, w slightly longer. Usually there s very little reason to prefer one term over any other, so don t spend a lot of time making your selection here.) Now, go through the list of these simple factors in your key term one at a time, determine the highest power to which each of them occurs in all (three, in this case) terms of the entire expression. This highest power will then be a common monomial factor of the entire expression. (Of course, if an item in this list doesn t occur at all in one of the terms, then this highest power will be the zero power that item is not a common factor of all of the terms in the expression.) So, for our example, we have five items in the list of potential common monomial factors to check. (i) 5: 25x 4 y 3 zw 2 contains x 3 y 3 zr 4 contains x 2 y 3 zw contains 5 1 The highest power of 5 common to all three terms (which is actually the lowest power that occurs in this list) is 5 1 = 5. Thus, 5 is a common factor of all three terms. (ii) x: 25x 4 y 3 zw 2 contains x 4 150x 3 y 3 zr 4 contains x 3 30x 2 y 3 zw contains x 2 Thus, the highest power of x common to all three terms is x 2, so x 2 is a common monomial factor of the entire expression. (iii) y: 25x 4 y 3 zw 2 contains y 3 150x 3 y 3 zr 4 contains y 3 30x 2 y 3 zw contains y 3 Thus, the highest power of y common to all three terms is y 3, so y 3 is a common monomial factor of the entire expression. (iv) z: 25x 4 y 3 zw 2 contains z 1 150x 3 y 3 zr 4 contains z 1 30x 2 y 3 zw contains z 1 Thus, the highest power of z common to all three terms is z 1 = z, so z is a common monomial factor of the entire expression. (v) w: 25x 4 y 3 zw 2 contains w 2 150x 3 y 3 zr 4 does not contain w (or contains w 0 ) 30x 2 y 3 zw contains w 1 Thus the highest power of w common to all three terms is w 0 = 1. This is the same thing as saying that there is no power of w common to all three terms, so there is no power of w which is a common monomial factor of all three terms. So, the common monomial factors of the entire three-term expression have been identified as 5, x 2, y 3, z. Our strategy has guaranteed that the product of these, 5x 2 y 3 z makes up the greatest monomial factor common to all three terms. Thus, going back to the template of our original goal, we can now write that 25x 4 y 3 zw x 3 y 3 zr x 2 y 3 zw = (5x 2 y 3 z)(whatever else) David W. Sabo (2003) Monomial Factors Page 4 of 7

5 All that needs determining yet is the form of the whatever else part. This we do in the same way as was done in the previous examples. Identify what is left of each term after the common monomial factor is removed. For the first term, 25x 4 y 3 zw 2 = 5 2 x 4 y 3 zw 2 = (5x 2 y 3 z)(5x 2 w 2 ) since to get 25x 4 y 3 zw 2 = 5 2 x 4 y 3 zw 2 from 5x 2 y 3 z, we need an additional factor of 5 (to make the 5 2 ), an additional factor of x 2 (to make the x 4 ), an additional factor of w 2 (to make the w 2 ). Multiplying the two factors in brackets on the right above is seen to regenerate the original term on the left. Now, repeat this process with each of the remaining two terms: 150x 3 y 3 zr 4 = x 3 y 3 zr 4 = (5x 2 y 3 z)(2 3 5xr 4 ) = (5x 2 y 3 z)(30xr 4 ) 30x 2 y 3 zw = 2 3 5x 2 y 3 zw = (5x 2 y 3 z)(2 3w) = (5x 2 y 3 z)(6w) So, 25x 4 y 3 zw x 3 y 3 zr x 2 y 3 zw = (5x 2 y 3 z)(5x 2 w xr 4 + 6w) This completes the operation of identifying the common monomial factors in the original expression. You should verify that multiplying to remove the brackets on the right-h side of this result gives precisely the expression on the left-h side, confirming that the two forms are mathematically equivalent. We have the required answer, but a couple more observations about this example may help you solve other problems of this sort. Notice that the second term in the example contained a power, r 4, of r. Yet our detailed analysis of potential monomial factors did not consider powers of r at all. We didn t consider powers of r, because r did not occur in the term 25x 4 y 3 zw 2 on which we were basing our analysis. But since r did not occur in this term at all, a power of r could not possibly be a common factor of all three terms. Because any common monomial factors must be a factor of the term on which we base our analysis, any symbols which do not appear in that term need not be considered at all. Hence the value in keying the analysis on a simpler-looking rather than more complicated-looking term in the original expression. Once you underst the strategy of the method displayed in great detail above, you may be able to accomplish the same end with much less writing without sacrificing the systematic approach. You still key the analysis on one of the terms in the expression, but common powers of factors are identified removed from the original expression in a stepwise fashion. 25x 4 y 3 zw x 3 y 3 zr x 2 y 3 zw 5 is a factor of the first term, each of the remaining terms contain a factor of 5, so remove a factor of 5 from all three terms. = 5(5x 4 y 3 zw x 3 y 3 zr 4 + 6x 2 y 3 zw) The only remaining numerical factor in the first term in the brackets is 5, but not all of the remaining terms in brackets contain a factor of 5, so there are no more numerical factors to remove. The first symbolic factor is a power of x. All three terms have a factor of at least x 2, so remove a factor of x 2 from the bracketed expression. David W. Sabo (2003) Monomial Factors Page 5 of 7

6 =5x 2 (5x 2 y 3 zw xy 3 zr 4 + 6y 3 zw) The next symbolic factor in the first term in the brackets is a power of y. All three terms contain a factor of y 3, so remove it. =5x 2 y 3 (5x 2 zw xzr 4 + 6zw) The next symbolic factor in the first term in the brackets is a z. Each of the three terms in brackets contains a factor z, so remove it. =5x 2 y 3 z(5x 2 w xr 4 + 6w) The next symbolic factor in the first term in brackets is a power of w. However not all of the remaining terms in the brackets contain a factor which is a power of w, so there is no power of w to be removed from the bracketed expression. Since we have run out of factors in the first term in brackets, we are done. The resultant expression has all common monomial factors identified. Notice that at each step, we needed to deal only with the expression left in the brackets, which is also generally getting simpler with each step (at least with those steps in which we ve been able to remove a factor). You can also check your work at each step because if you remultiply by the most recently removed factor, you must obtain the expression from the previous step. This has been a rather lengthy discussion of issues approaches to factoring one fairly complicated expression. To conclude this document, we ll briefly demonstrate the identification of monomial factors for a couple of additional expressions. Example 5: Remove all common monomial factors from the expression 3r 4 s 3 + 6r 3 s 4 105r 2 s 5 The first term in this expression has factors of 3, r, s. So, we need to check whether 3 is a common factor in all three terms, whether there are common powers of r s in all three terms. 3r 4 s 3 + 6r 3 s 4 105r 2 s 5 3 is a factor of the first term, upon inspection is found to be a factor of the remaining two terms as well. So, remove a factor of 3 from all three terms. = 3(r 4 s 3 + 2r 3 s 4 35r 2 s 5 ) Each term contains a factor of r 2. Remove it. = 3r 2 (r 2 s 3 + 2rs 4 35s 5 ) Each term in the brackets now contains a factor of s 3. Remove it. = 3r 2 s 3 (r 2 + 2rs 35s 2 ) Since the first term in brackets contains only powers of r, but the third term contains no factor of r, there cannot be any further common monomial factors in the three terms of the expression in brackets. The required factorization of monomial factors is thus complete. Thus, with all fo the common monomial factors identified, we can write the final result as: 3r 4 s 3 + 6r 3 s 4 105r 2 s 5 = 3r 2 s 3 (r 2 + 2rs 35s 2 ) David W. Sabo (2003) Monomial Factors Page 6 of 7

7 (Once you ve read the next section in these notes, you ll be able to see that the expression in brackets in the final answer of this last example can, in fact, be written as a product, though not of monomial factors. So, this is a final answer only as far as the methods presented so far here are concerned.) Example 6: Remove all common monomial factors from the expression 54x 3 y 2 z 150xy 4 z Looking at the first term here, we see that in addition to a possible common numerical factor, we need to check for common powers of x, y, z in the two terms. The hardest part of this example is in determining the common numerical factor in the two terms. However, writing the two numerical coefficients as products of prime factors (we described how to do this in the note called Prime Factors in the section on numerical fractions), we get 54 = 2 x 3 x 3 x = 2 x 3 x 5 x 5 Thus, both terms share a factor of 2 x 3 = 6. Proceeding as in previous examples, we now get: 54x 3 y 2 z 150xy 4 z = 6(9x 3 y 2 z 25xy 4 z) removing the already identified common numerical factor of 6. Now we note that both terms share a factor of x 1 = x, so remove this factor. = 6x(9x 2 y 2 z 25y 4 z) Next, there is a common factor of y 2 shared by the two terms, so remove it. = 6xy 2 (9x 2 z 25y 2 z) Finally, both terms share a factor of z, so remove it. = 6xy 2 z(9x 2 25y 2 ) Thus, the final answer here is 54x 3 y 2 z 150xy 4 z = 6xy 2 z(9x 2 25y 2 ) Shortly, we ll demonstrate that the bracketed expression in the answer given above in Example 6 can be factored into the product of two binomials, so that the most factored form of the original expression is 54x 3 y 2 z 150xy 4 z = 6xy 2 z(3x + 5y)(3x 5y). However, the identification of all common monomial factors is the necessary first step before more specialized types of factorization can be attempted. David W. Sabo (2003) Monomial Factors Page 7 of 7

### 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,

### 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

### 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.

### A 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

### 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.

### 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 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 Trinomials of the Form x 2 bx c

4.2 Factoring Trinomials of the Form x 2 bx c 4.2 OBJECTIVES 1. Factor a trinomial of the form x 2 bx c 2. Factor a trinomial containing a common factor NOTE The process used to factor here is frequently

### 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

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

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

### 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

### 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

### 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

### Using 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 trial-and-error

### 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

### 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

### 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,

### 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

### 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,...}

### Section 1. Finding Common Terms

Worksheet 2.1 Factors of Algebraic Expressions Section 1 Finding Common Terms In worksheet 1.2 we talked about factors of whole numbers. Remember, if a b = ab then a is a factor of ab and b is a factor

### 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 A Quadratic Polynomial If we multiply two binomials together, the result is a quadratic polynomial: This multiplication is pretty straightforward, using the distributive property of multiplication

### 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".

### How To Solve Factoring Problems

05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring

### Factoring - Greatest Common Factor

6.1 Factoring - Greatest Common Factor Objective: Find the greatest common factor of a polynomial and factor it out of the expression. The opposite of multiplying polynomials together is factoring polynomials.

### 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/

### 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 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

### 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

### 1.4. 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.

### 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

### 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

### BEGINNING ALGEBRA ACKNOWLEDMENTS

BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science

### 1.6 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

### Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF

Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials

### Polynomials and Factoring. Unit Lesson Plan

Polynomials and Factoring Unit Lesson Plan By: David Harris University of North Carolina Chapel Hill Math 410 Dr. Thomas, M D. 2 Abstract This paper will discuss, and give, lesson plans for all the topics

### 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 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

### 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

### ( ) 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

### Simplifying Algebraic Fractions

5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions

### Introduction to Fractions

Section 0.6 Contents: Vocabulary of Fractions A Fraction as division Undefined Values First Rules of Fractions Equivalent Fractions Building Up Fractions VOCABULARY OF FRACTIONS Simplifying Fractions Multiplying

### 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

### CAHSEE on Target UC Davis, School and University Partnerships

UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

### 5.1 FACTORING OUT COMMON FACTORS

C H A P T E R 5 Factoring he sport of skydiving was born in the 1930s soon after the military began using parachutes as a means of deploying troops. T Today, skydiving is a popular sport around the world.

### In the above, the number 19 is an example of a number because its only positive factors are one and itself.

Math 100 Greatest Common Factor and Factoring by Grouping (Review) Factoring Definition: A factor is a number, variable, monomial, or polynomial which is multiplied by another number, variable, monomial,

### 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

### 0.8 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

### 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

### 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

### Vocabulary 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

### 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

### 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!

### Factoring Polynomials

Factoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations

### a. You can t do the simple trick of finding two integers that multiply to give 6 and add to give 5 because the a (a = 4) is not equal to one.

FACTORING TRINOMIALS USING THE AC METHOD. Factoring trinomial epressions in one unknown is an important skill necessary to eventually solve quadratic equations. Trinomial epressions are of the form a 2

### 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

### Algebraic 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

### 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.

### 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

### Negative Integer Exponents

7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions

### Chapter 3 Section 6 Lesson Polynomials

Chapter Section 6 Lesson Polynomials Introduction This lesson introduces polynomials and like terms. As we learned earlier, a monomial is a constant, a variable, or the product of constants and variables.

### SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

### 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

### 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

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### 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

### 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

### 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.

### 3.1. 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

### 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 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 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...

### A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

### 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.

### 6.3 FACTORING ax 2 bx c WITH a 1

290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100

### 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.

### Simplification Problems to Prepare for Calculus

Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.

### 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

### 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

### 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?

### 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)

### Prime Factorization, Greatest Common Factor (GCF), and Least Common Multiple (LCM)

Prime Factorization, Greatest Common Factor (GCF), and Least Common Multiple (LCM) Definition of a Prime Number A prime number is a whole number greater than 1 AND can only be divided evenly by 1 and itself.

### Solution Guide Chapter 14 Mixing Fractions, Decimals, and Percents Together

Solution Guide Chapter 4 Mixing Fractions, Decimals, and Percents Together Doing the Math from p. 80 2. 0.72 9 =? 0.08 To change it to decimal, we can tip it over and divide: 9 0.72 To make 0.72 into a

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### Welcome 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

### Polynomial Expression

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

### 1.4. 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

### 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

### 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

### FACTORISATION 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

### 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

### 7-6. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content

7-6 Choosing a Factoring Model Extension: Factoring Polynomials with More Than One Variable Essential question: How can you factor polynomials with more than one variable? What is the connection between

### Factor Polynomials Completely

9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping

### Five 5. Rational Expressions and Equations C H A P T E R

Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.

### 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