Common Factors

11 Key excerpts on "Common Factors"

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  eBook - PDF

  Years 9 - 10 Maths For Students

  • (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.

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  Introductory Mathematics

  • J Daniels, M Kropman, J Daniels, M Kropman(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  • The lowest common multiple ( LCM ) of two or more numbers is the smallest possible number into which all the numbers will divide exactly. • Factorising Taking out the highest common factor. Common Factors are the factors shared by two or more numbers. A common factor can be a number or a variable, or a combination of numbers and variables, for example: 4 x 2 y 3 z 2 + 16 x 3 y 4 z 2 – 64 xyz 3 = 4 xyz 2 ( xy 2 + 4 x 2 y 3 – 16 z ) Grouping An expression with more than three terms can be factorised by grouping terms to find a common factor, for example: ax + ay – bx – by = ( ax + ay ) + (– bx – by ) = a ( x + y ) + b (– x – y ) = a ( x + y ) – b ( x + y ) = ( x + y )( a + b ) • Algebraic fractions Rules of fractions Rule Explanation 1 a b × c d = ac bd Also: a × c d = a 1 × c d = ac d When multiplying fractions, multiply all the numerators and then all the denominators . Restrictions: b ≠ 0; d ≠ 0 105 Introductory Mathematics| Hands-On Rule Explanation 2 a b ÷ c d = a b × d c = ad bc When dividing fractions, change the operation from division to multiplication and invert whatever is after the division sign (multiplying with the reciprocal of the fraction). This is called the tips and times rule. Restrictions: b ≠ 0; c ≠ 0; d ≠ 0 3 a b + c b = a c b + or a b + c d = ad bc bd + To add or subtract fractions, the denominator must be the same . Restrictions: b ≠ 0; d ≠ 0 • Multiplying and dividing fractions Steps for multiplying fractions Step 1 Cancel factors in the numerator with factors in the denominator. Step 2 Multiply the numerators by numerators. Step 3 Multiply the denominators by denominators. Step 4 Simplify the answer to its simplest form. Steps for dividing fractions Step 1 First change the ÷ (division) sign to a × (multiplication) sign, and invert the fraction after the division sign. Step 2 Follow the steps above for multiplying fractions.

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  Algebra I For Dummies

  • Mary Jane Sterling(Author)
  • 2016(Publication Date)
  • For Dummies

    (Publisher)

  Pulling out factors and leaving the rest

  Pulling out Common Factors from lists of terms or the sums or differences of a bunch of terms is done for a good reason. It’s a common task when you’re simplifying expressions and solving equations. The common factor that makes the biggest difference in these problems is the greatest common factor (GCF). When you recognize the GCF and factor it out, it does the most good.

  The greatest common factor is the largest-possible number that evenly divides each term of an expression containing two or more terms (or evenly divides the numerator and denominator of a fraction).

  In any factoring discussion, the GCF, the most common and easiest factoring method, always comes up first. And it’s helpful to know about the GCF when solving equations. In an expression with two or more terms, finding the greatest common factor can make the expression more understandable and manageable.

  When simplifying expressions, the best-case scenario is to recognize and pull out the GCF from a list of terms. Sometimes, though, the GCF may not be so recognizable. It may have some strange factors, such as 7, 13, or 23. It isn’t the end of the world if you don’t recognize one of these numbers as being a multiplier; it’s just nicer if you do.

  The three terms in the expression 12x 2 y 4 + 16xy 3 – 20x 3 y 2

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  N1 Mathematics

  • J Daniels, M Kropman, J Daniels, M Kropman(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  • The lowest common multiple ( LCM ) of two or more numbers is the smallest possible number into which all the numbers will divide exactly. • Factorising Taking out the highest common factor. Common Factors are the factors shared by two or more numbers. A common factor can be a number or a variable, or a combination of numbers and variables, for example: 4 x 2 y 3 z 2 + 16 x 3 y 4 z 2 – 64 xyz 3 = 4 xyz 2 ( xy 2 + 4 x 2 y 3 – 16 z ) Grouping An expression with more than three terms can be factorised by grouping terms to find a common factor, for example: ax + ay – bx – by = ( ax + ay ) + (– bx – by ) = a ( x + y ) + b (– x – y ) = a ( x + y ) – b ( x + y ) = ( x + y )( a + b ) • Algebraic fractions Rules of fractions Rule Explanation 1 a b × c d = ac bd Also: a × c d = a 1 × c d = ac d When multiplying fractions, multiply all the numerators and then all the denominators . Restrictions: b ≠ 0; d ≠ 0 105 N1 Mathematics| Hands-On Rule Explanation 2 a b ÷ c d = a b × d c = ad bc When dividing fractions, change the operation from division to multiplication and invert whatever is after the division sign (multiplying with the reciprocal of the fraction). This is called the tips and times rule. Restrictions: b ≠ 0; c ≠ 0; d ≠ 0 3 a b + c b = a c b + or a b + c d = ad bc bd + To add or subtract fractions, the denominator must be the same . Restrictions: b ≠ 0; d ≠ 0 • Multiplying and dividing fractions Steps for multiplying fractions Step 1 Cancel factors in the numerator with factors in the denominator. Step 2 Multiply the numerators by numerators. Step 3 Multiply the denominators by denominators. Step 4 Simplify the answer to its simplest form. Steps for dividing fractions Step 1 First change the ÷ (division) sign to a × (multiplication) sign, and invert the fraction after the division sign. Step 2 Follow the steps above for multiplying fractions.

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  Elementary Algebra

  • Toby Wagner(Author)
  • 2021(Publication Date)
  • Chemeketa Press

    (Publisher)

  B. Factoring Out the Greatest Common Factor

  In the previous examples, we knew the product and one of the factors, so we simply used division to figure out the other factor. But suppose we’re given a polynomial product without any factors. We need to be able to determine a monomial that is a factor of each of the terms in the polynomial. If such a monomial exists, it’s called a

  common factor

  . In the expression 8x + 12, for example, we know that 2 is a factor of 8x and of 12. We can thus use 2 as one factor and use division to find the other factor: = 4x + 6. We now rewrite 8x + 12 as 2(4x + 6). This is called factoring out a common factor.

  The Greatest Common Factor

  Whenever we factor out a monomial from a polynomial, we try to find a monomial that is more than just a common factor of each term of the polynomial. We are looking for the greatest common factor of the terms in the polynomial.

  The

  greatest common factor

  (GCF) of two numbers is the largest number that is a factor of each. For example, consider the numbers 8 and 12. The factors of 8 are 1, 2, 4, and 8. The factors of 12 are 1, 2, 3, 4, 6, and 12. The Common Factors are 1, 2, and 4. The greatest of these, 4, is the GCF of 8 and 12.

  When dealing with a polynomial that has multiple terms, the GCF of theses terms has the following qualities: 1.  The numerical coefficient is the GCF of the numerical coefficients of the terms of the polynomial.

  2.  The variables represent the lowest powers of the variables in the terms of the polynomial.

  The rule pertaining to variables might seem counterintuitive, so let’s consider x4 and x2 . The factors of x4 are 1, x, x2 , x3 , and x4 . The factors of x7 are 1, x, x2 , x3 , x4 , x5 , x6 , and x7 . When we compare these lists of factors, we see that x4 is the GCF of x4 and x7 .

  To factor out the GCF, we first must determine what the GCF is and write it. Then we divide each term of the original polynomial by the GCF and write the resulting polynomial in parentheses.

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  Introductory 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.

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  Beginning Algebra

  Connecting Concepts through Applications

  • Mark Clark, Cynthia Anfinson(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  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. Factor a Polynomial To factor a polynomial means to rewrite the polynomial expression as the product of simpler polynomials that, when multiplied together, result in the original polynomial. When factoring an expression, it is best to begin by looking for the greatest common factor , or GCF, which helps to break down all the terms. DEFINITION Greatest Common Factor (GCF) of Integers or Monomials The greatest common factor, or GCF, of a set of integers or monomials is the largest factor that is shared in common with all of the elements of the set. To find the GCF of a set of integers or monomials, use the following steps. Steps to Find the GCF of a Set of Integers or Monomials 1. Find the prime factorization of each integer or coefficient. Write repeated factors in exponential form. 2. Circle the Common Factors of all the elements in the set. 3. The GCF equals the product of the Common Factors, each raised to the lowest power that appears. 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.

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  Mathematical Practices, Mathematics for Teachers

  Activities, Models, and Real-Life Examples

  • Ron Larson, Robyn Silbey(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The notation for the greatest common factor of a and b is written as GCF(a, b). To find the GCF(a, b), write the prime factorizations of a and b. The greatest common factor is the product of the Common Factors. Example a = 20, b = 12 a = 20 b = 12 2 2 5 3 Common Factors a = 2 ⋅ 2 ⋅ 5 b = 2 ⋅ 2 ⋅ 3 GCF(20, 12) = 2 ⋅ 2 = 4 If a divides b, then GCF(a, b) = a. If b divides a, then GCF(a, b) = b. Example GCF(3, 12) = 3 EXAMPLE 1 Finding the Greatest Common Factor Find the greatest common factor of 16 and 24. SOLUTION Begin by writing the prime factorization of each number. a = 16: 16 b = 24: 24 4 4 4 6 2 2 2 2 2 2 2 3 So, 16 = 2 ⋅ 2 ⋅ 2 ⋅ 2 and 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3. Next, use a diagram to organize the prime factors of a and b. a = 16 b = 24 2 2 2 2 3 Common Factors The greatest common factor of 16 and 24 is 2 ⋅ 2 ⋅ 2 = 8. When you are teaching strategies for finding the GCF of two whole numbers, remind your students to first think about the numbers. 1. If both are primes, then the GCF is 1. GCF(3, 5) = 1 2. If one number is a multiple of the other, then the GCF is the lesser of the two numbers. GCF(3, 15) = 3 3. If one of the numbers is 0, then the GCF is the greater of the two numbers because 0 is divisible by any nonzero number. GCF(3, 0) = 3 Classroom Tip Standards Grades 3–5 Operations and Algebraic Thinking Students should gain familiarity with factors and multiples. Grades 6–8 The Number System Students should find Common Factors and multiples. Copyright 2014 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.

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  Technical Mathematics with Calculus

  • Paul A. Calter, Michael A. Calter(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  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. See if you get the same results. (a) (b) (c) (d) ◆◆◆ 3x 3  2x  5x 4  x(3x 2  2  5x 3 ) 3x 3  6x 2 y  9x 4 y 2  3x 2 (x  2y  3x 2 y 2 ) 3xy 2  9x 3 y  6x 2 y 2  3xy(y  3x 2  2xy) 2x 2  x  x(2x  1) x 3  3x  x (x 2  3) Factors in the Denominator Common Factors may appear in the denominators of the terms as well as in the numerators. ◆◆◆ Example 4: Here we show some examples of the factoring out of Common Factors from the denominators. (a) (b) ◆◆◆ Checking To check if factoring has been done correctly, simply multiply your factors together and see if you get back the original expression. This check will tell whether you have factored correctly, but not whether you have factored completely. ◆◆◆ Example 5: Let us check Example 4(b) by multiplying out. Checks. ◆◆◆  x y 2  x 2 y  2x 3y x y a 1 y  x  2 3 b  a x y b a 1 y b  a x y b x  a x y b a 2 3 b x y 2  x 2 y  2x 3y  x y a 1 y  x  2 3 b 1 x  2 x 2  1 x a 1  2 x b Students are sometimes puzzled over the “1” in Example 3(a). Why should it be there? After all, when you remove a chair from a room, it is gone; there is nothing (zero) remaining where the chair used to be. If you remove an x by factoring, you might assume that nothing (zero) remains where the x used to be. Prove to yourself that this is not correct by multiplying the factors to see if you get back the original expression. 2x 2  x  x(2x  0)? Common Error 322 Chapter 11 ◆ Factoring and Fractions Factoring by Calculator A calculator that can manipulate algebraic symbols can be used to factor an expres- sion. We will do a simple factoring here, with more complex types to come later. ◆◆◆ Example 6: Factor the expression from Example 3(a): on the TI-89 or similar calculator.

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  The Higher Arithmetic

  An Introduction to the Theory of Numbers

  • H. Davenport(Author)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

  These considerations extend easily to more than two numbers. But then it is important to note the two kinds of relative primality that are possible. Several numbers are said to be relatively prime if there is no number greater than 1 which divides all of them; they are said to be relatively prime in pairs if no two of them have a common factor greater than 1. The condition for the former is that there shall be no one prime occurring in all the numbers, and for the latter it is that there shall be no prime that occurs in any two of the numbers. There are several simple theorems on divisibility which one is inclined to think of as obvious, but which are in fact only obvious in the light of the uniqueness of factorization into primes. For example, if a number divides the product of two numbers and is relatively prime to one of them it must divide the other. If a divides bc and is relatively prime to b, the prime 16 The Higher Arithmetic factorization of a is contained in that of bc but has nothing in common with that of b, and is therefore contained in that of c. 6. Euclid’s algorithm In Prop. 2 of Book VII, Euclid gave a systematic process, or algorithm, for finding the highest common factor of two given numbers. This algorithm provides a different approach to questions of divisibility from that adopted in the last two sections, and we therefore begin again without assuming anything except the mere definition of divisibility. Let a and b be two given natural numbers, and suppose that a > b. We propose to investigate the common divisors of a and b. If a is divisible by b, then the common divisors of a and b consist simply of all divisors of b, and there is no more to be said. If a is not divisible by b, we can express a as a multiple of b together with a remainder less than b, that is a = qb + c, where c < b.

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  Elementary Algebra 2e

  • Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
  • 2020(Publication Date)
  • Openstax

    (Publisher)

  Find the Greatest Common Factor of Two or More Expressions Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring. We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM. Greatest Common Factor The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions. First we’ll find the GCF of two numbers. EXAMPLE 7.1 HOW TO FIND THE GREATEST COMMON FACTOR OF TWO OR MORE EXPRESSIONS Find the GCF of 54 and 36. Solution 808 Chapter 7 Factoring This OpenStax book is available for free at http://cnx.org/content/col31130/1.4 Notice that, because the GCF is a factor of both numbers, 54 and 36 can be written as multiples of 18. 54 = 18 · 3 36 = 18 · 2 TRY IT : : 7.1 Find the GCF of 48 and 80. TRY IT : : 7.2 Find the GCF of 18 and 40. We summarize the steps we use to find the GCF below. In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor. EXAMPLE 7.2 Find the greatest common factor of 27x 3 and 18x 4 . Solution Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the Common Factors in each column. Bring down the Common Factors. Multiply the factors. The GCF of 27x 3 and 18x 4 is 9x 3 . TRY IT : : 7.3 Find the GCF: 12x 2 , 18x 3 . TRY IT : : 7.4 Find the GCF: 16y 2 , 24y 3 . EXAMPLE 7.3 Find the GCF of 4x 2 y, 6xy 3 . HOW TO : : FIND THE GREATEST COMMON FACTOR (GCF) OF TWO EXPRESSIONS. Factor each coefficient into primes. Write all variables with exponents in expanded form. List all factors—matching Common Factors in a column.

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