Mathematics
Factors
In mathematics, factors are numbers that can be multiplied together to produce a given number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Understanding factors is important in various mathematical operations, such as finding the greatest common factor or simplifying fractions.
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8 Key excerpts on "Factors"
eBook - PDF- (Author)
- 2015(Publication Date)
- For Dummies(Publisher)
Fractions reduce more easily, equations solve more easily, and answers are observed more easily when you can factor. 180 Part II: Algebra is Part of Everything Factoring out numbers Factoring is the opposite of distributing; it’s ‘undistributing’ (refer to Chapter 7 for more on distribution). When performing distribution, you multiply a series of terms by a common multiplier. Now, by factoring, you seek to find what a series of terms has in common and then take it away, dividing the common factor or multiplier out from each term. Think of each term as a numerator of a fraction, and you’re finding the same denominator for each. By factoring out, the common factor is put outside parentheses or brackets and all the results of the divisions are left inside. An expression can be written as the product of the largest number that divides all the terms evenly times the results of the divisions: ab + ac + ad = a(b + c + d ). Writing factoring as division In the trinomial 16a − 8b + 40c 2 , 2 is a common factor. But 4 is also a common factor, and 8 is a common factor. Here are the divisions of the terms by 2, 4 and 8: Reviewing the terms and rules You’ll understand factoring better if you have a firm handle on what the terms used to talk about factoring mean: 6 Term: A group of number(s) and/or varia- ble(s) connected to one another by multi- plication or division and separated from other terms by addition or subtraction. 6 Factor: Any of the values involved in a multiplication problem that, when multi- plied together, produce a result. 6 Coefficient: A number that multiplies a variable and tells how many of the variable. 6 Constant: A number or variable that never changes in value. 6 Relatively prime: Terms that have no Factors in common. If the only factor that numbers share in common is 1, they’re considered relatively prime. Here is an illustration for all the terms I just gave: In the expression 5xy + 4z − 6, you see three terms.
eBook - PDFBeginning Algebra
Connecting Concepts through Applications
- Mark Clark, Cynthia Anfinson(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
531 © iStock.com/simonkr S tudents of business and management analyze profit, revenue, and cost equations. These equations are often quadratic equations . For example, an appliance manufacturer may want to know how many appliances they have to sell in order to maintain a certain level of average profit per appliance. In this chapter, we begin our study of how to factor quadratic expressions and solve quadratic equations. Factoring 6 6.1 What It Means to Factor 6.2 Factoring Trinomials 6.3 Factoring Special Forms 6.4 Solving Quadratic Equations by Factoring Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. C H A P T E R 6 F a c t o r i n g 532 In Section R.2, we discussed factoring numbers. Recall that when asked to factor a number, that means to rewrite that number as a product. For example, the prime Factors of 10 are 5 and 2 because 5 # 2 5 10 . Factoring 10 means to rewrite it as the product (multiplication) of its Factors, 5 and 2. Instructions What to do Multiply 5 and 2. Given 5 and 2, multiply them to get 10. 5 # 2 5 10 Factor 10. Given 10, find the Factors that multiply to 10. 10 5 5 # 2 What It Means to Factor LEARNING OBJECTIVES Find the greatest common factor. Factor out the greatest common factor. Factor by grouping. Explain how to factor completely. 6.1 DEFINITIONS Factor an Integer To factor an integer means to rewrite it as the product of other integers that, when multiplied together, result in the original integer.
eBook - PDF- Dale Ewen(Author)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
326 CHAPTER 10 ◆ Factoring Algebraic Expressions Finding Monomial Factors 10.1 10.1 Factoring an algebraic expression , like finding the prime Factors of a number, means writing the expression as a product of Factors . The prime factorization of 12 is 2 ? 2 ? 3. Other factorizations of 12 are 2 ? 6 and 4 ? 3 . Since factorization means writing a number or an algebraic expression as a product, then a number or an algebraic expression divided by one factor generates another factor. Thus, 12 divided by 2 gives 6, so 2 times 6 is a fac-torization of 12. To factor the expression 2 x 1 2 y , notice that 2 is a factor common to both terms of the expression. In other words, 2 is a factor of 2 x 1 2 y . To find the other factor, divide by 2. 2 x 1 2 y 2 5 2 x 2 1 2 y 2 5 x 1 y Therefore, a factorization of 2 x 1 2 y is 2( x 1 y ). A monomial factor is a one-term factor that divides each term of an algebraic expres-sion. Here, 2 divides each term of the algebraic expression and is called a monomial factor. When factoring any algebraic expression, always look first for monomial Factors that are common to all terms. Factor: 3 a 1 6 b . First, look for a common monomial factor. Since 3 divides both 3 a and 6 b , 3 is a common monomial factor of 3 a 1 6 b . Divide 3 a 1 6 b by 3. 3 a 1 6 b 3 5 3 a 3 1 6 b 3 5 a 1 2 b Thus, 3 a 1 6 b 5 3( a 1 2 b ). Check this result by multiplication: 3( a 1 2 b ) 5 3 a 1 6 b . ◆ Factor: 4 x 2 1 8 x 1 12. Since 4 divides each term of the expression, divide 4 x 2 1 8 x 1 12 by 4 to obtain the other factor. 4 x 2 1 8 x 1 12 4 5 4 x 2 4 1 8 x 4 1 12 4 5 x 2 1 2 x 1 3 Thus, 4 x 2 1 8 x 1 12 5 4( x 2 1 2 x 1 3). 4( x 2 1 2 x 1 3) 5 4 x 2 1 8 x 1 12 Your product should be the original expression. NOTE: In this example, 2 is also a common factor of each term of the expression. However, 4 is the greatest common factor. The greatest common factor of a polynomial is the largest common factor that divides all terms in the expression.
eBook - PDFIntroductory Algebra
Concepts with Applications
- Charles P. McKeague(Author)
- 2013(Publication Date)
- XYZ Textbooks(Publisher)
Note As you will see as we progress through the book, factoring is a tool that is used in solving a number of problems. Before seeing how it is used, however, we first must learn how to do it. So, in this section and the sections that follow, we will be developing our factoring skills. DEFINITION greatest common factor The greatest common factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. Image © sxc.hu, Kaliyoda, 2009 KEY WORDS 385 5.1 The Greatest Common Factor and Factoring by Grouping EXAMPLE 1 Find the greatest common factor for the polynomial 3x 5 + 12x 2 . Solution The terms of the polynomial are 3x 5 and 12x 2 . The largest number that divides the coefficients is 3, and the highest power of x that is a factor of x 5 and x 2 is x 2 . Therefore, the greatest common factor for 3x 5 + 12x 2 is 3x 2 ; that is, 3x 2 is the largest monomial that divides each term of 3x 5 + 12x 2 . EXAMPLE 2 Find the greatest common factor for 8a 3 b 2 + 16a 2 b 3 + 20a 3 b 3 . Solution The largest number that divides each of the coefficients is 4. The highest power of the variable that is a factor of a 3 b 2 , a 2 b 3 , and a 3 b 3 is a 2 b 2 . The greatest common factor for 8a 3 b 2 + 16a 2 b 3 + 20a 3 b 3 is 4a 2 b 2 . It is the largest monomial that is a factor of each term. Once we have recognized the greatest common factor of a polynomial, we can apply the distributive property and factor it out of each term. We rewrite the polynomial as the product of its greatest common factor with the polynomial that remains after the greatest common factor has been factored from each term in the original polynomial. EXAMPLE 3 Factor the greatest common factor from 3x − 15. Solution The greatest common factor for the terms 3x and 15 is 3. We can rewrite both 3x and 15 so that the greatest common factor 3 is showing in each term. It is important to realize that 3x means 3 ⋅ x. The 3 and the x are not “stuck” together.
eBook - PDFAlgebra
A Combined Course 2E
- Charles P. McKeague(Author)
- 2018(Publication Date)
- XYZ Textbooks(Publisher)
What skills do you have that align with those of an independent learner? What attributes do you have that keep you from being an independent learner? What qualities would you like to obtain that you don't have now? 449 6.1 A Factor the greatest common factor from a polynomial. B Factor polynomials by grouping. OBJECTIVES greatest common factor largest monomial factoring by grouping KEY WORDS THE GREATEST COMMON FACTOR AND FACTORING BY GROUPING 6.1 The Greatest Common Factor and Factoring by Grouping © istockphoto.com/rihardzz In flight archery, a person uses a bow to shoot an arrow into the air with a goal of achieving the greatest distance, as opposed to aiming for a specific target. The curved path that the arrow takes from the bow to the ground creates the shape of a parabola, which we will discuss further in Chapter 10. For now, let’s say the arrow’s path can be represented by the polynomial −16x 2 + 62x + 8. To begin factoring this polynomial, we need to find the greatest common factor, which is the primary focus of this section. Recall the following diagram to illustrate the relationship between multiplication and factoring. Multiplication Factors → 3 ⋅ 5 = 15 ← Product Factoring A similar relationship holds for multiplication of polynomials. Reading the following diagram from left to right, we say the product of the binomials x + 2 and x + 3 is the trinomial x 2 + 5x + 6. However, if we read in the other direction, we can say that x 2 + 5x + 6 Factors into the product of x + 2 and x + 3. Multiplication Factors → (x + 2)(x + 3) = x 2 + 5x + 6 ← Product Factoring In this chapter, we develop a systematic method of factoring polynomials. A Factoring The Greatest Common Factor In this section, we will apply the distributive property to polynomials to factor from them what is called the greatest common factor .
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
The distance x to the center of gravity is given by Could you solve this equation for, say, ? In this chapter we show how to solve this type of equation. m 1 x 10m 1 25m 2 m 1 m 2 320 Chapter 11 ◆ Factoring and Fractions Here we will show traditional pencil-and-paper methods of solution alongside solutions by a calculator such as the TI-89 that can manipulate algebraic expressions. 11–1 Common Factors As usual, we must start with definitions of the terms we will be using in the following sections. Factors of an Expression The Factors of an expression are those quantities whose product is the original expression. ◆◆◆ Example 1: The Factors of are and , because ◆◆◆ Many expressions have no Factors other than 1 and themselves. Such expres- sions are called prime. Factoring is the process of finding the Factors of an expression. It is the reverse of finding the product of two or more quantities. We usually factor an expression by recognizing the form of that expression. In the first type of factoring we will cover, we look for common Factors. Common Factors ■ Exploration: Try this. Expand this expression by multiplying out. Now how would you return the expression you just got back to its original form? Can you state your findings as a general rule? ■ If each term of an expression contains the same quantity (called the common factor), the quantity may be factored out. Common Factor 10 This is nothing but the distributive law that we studied earlier. ◆◆◆ Example 2: In the expression x 3 3x ab ac a(b c) a(b c d) x 2 4x x(x 4) Factoring (Finding the Factors) x(x 4) x 2 4x Multiplication (Finding the Product) x 2 9 (x 3)(x 3) x 2 3x 3x 9 x 3 x 3 x 2 9 Section 1 ◆ Common Factors 321 each term contains an x as a common factor. So we write Most of the factoring we will do will be of this type. ◆◆◆ ◆◆◆ Example 3: Here are some examples of the factoring out of common Factors.
eBook - PDFMaths: A Student's Survival Guide
A Self-Help Workbook for Science and Engineering Students
- Jenny Olive(Author)
- 2003(Publication Date)
- Cambridge University Press(Publisher)
1 Basic algebra: some reminders of how it works In many areas of science and engineering, information can be made clearer and more helpful if it is thought of in a mathematical way. Because this is so, algebra is extremely important since it gives you a powerful and concise way of handling information to solve problems. This means that you need to be confident and comfortable with the various techniques for handling expressions and equations. The chapter is divided up into the following sections. 1.A Handling unknown quantities (a) Where do you start? Self-test 1, (b) A mind-reading explained, (c) Some basic rules, (d) Working out in the right order, (e) Using negative numbers, (f) Putting into brackets, or factorising 1.B Multiplications and factorising: the next stage (a) Self-test 2, (b) Multiplying out two brackets, (c) More factorisation: putting things back into brackets 1.C Using fractions (a) Equivalent fractions and cancelling down, (b) Tidying up more complicated fractions, (c) Adding fractions in arithmetic and algebra, (d) Repeated Factors in adding fractions, (e) Subtracting fractions, (f) Multiplying fractions, (g) Dividing fractions 1.D The three rules for working with powers (a) Handling powers which are whole numbers, (b) Some special cases 1.E The different kinds of numbers (a) The counting numbers and zero, (b) Including negative numbers: the set of integers, (c) Including fractions: the set of rational numbers, (d) Including everything on the number line: the set of real numbers, (e) Complex numbers: a very brief forwards look 1.F Working with different kinds of number: some examples (a) Other number bases: the binary system, (b) Prime numbers and Factors, (c) A useful application – simplifying square roots, (d) Simplifying fractions with signs underneath 1.A Handling unknown quantities 1.A. (a) Where do you start? Self-test 1 All the maths in this book which is directly concerned with your courses depends on a foundation of basic algebra.
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
3 MODULE Factorisation and fractions 3.1 Factorisation, HCF and LCM On completion of this module, you should be able to: 3.1.1 Find the Factors of an expression 3.1.2 Determine the HCF (highest common factor) 3.1.3 Determine the LCM (lowest common multiple) 3.1.4 Factorise polynomials with common Factors, and regroup terms with a common factor. 3.2 Algebraic fractions On completion of this module, you should be able to: 3.2.1 Simplify fractions 3.2.2 Multiply and divide fractions 3.2.3 Add and subtract algebraic fractions. 76 Module 3 • Factorisation and fractions 3.1 Factorisation, HCF and LCM Introduction • The process of writing a mathematical statement as a product of its Factors is called factorisation . • Factorisation of expressions is the reverse process of developing products (expanding). • The Factors of an expression are two or more expressions that can be multiplied to produce the original expression. Example ax + bx + cx = x ( a + b + c ) • x and ( a + b + c ) are Factors of all three terms, ax + bx + cx . By expanding: x ( a + b + c ) = ax + bx + cx Consider the following. multiplication x ( a + b + c ) = ax + bx + cx factorisation Pre-knowledge • A monomial is a single algebraic term, for example 3 x 2 y 3 z or 2( x + y ). • Polynomials are algebraic expressions with two or more terms that involve numbers and variables (symbols such as x , y , z ). The variables must have positive integer exponents and can only appear in the numerator of the expression. Example Consider the polynomial: 2 x 3 – 4 x 2 + 3 x – 1 The degree of the expression is 3 (highest power of x in the expression). The coefficient of x 2 is –4. The exponent (index) of the variable in the third term is 1. The constant term is –1. The variable in the expression is x. There are 4 terms. The polynomial is written in descending powers of x. • 2 x 3 – 4 x 2 + 3 x – 1 is a polynomial, but 2 8 x + 12 x 6 – x 1 2 is not a polynomial.
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