Mathematics
Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into simpler components, typically by finding its factors. This process is important in algebra and calculus, as it helps solve equations, find roots, and simplify expressions. Factoring can be done using various methods, such as the distributive property, grouping, difference of squares, and more.
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eBook - PDFIntroductory Algebra
Concepts with Applications
- Charles P. McKeague(Author)
- 2013(Publication Date)
- XYZ Textbooks(Publisher)
This section is important because it will give you an opportunity to factor a variety of polynomials such as the one above. Prior to this section, the polynomials you worked with were grouped together according to the method used to factor them; that is, in Section 5.4 all the polynomials you factored were either the difference of two squares or perfect square trinomials. What usually happens in a situation like this is that you become proficient at factoring the kind of polynomial you are working with at the time but have trouble when given a variety of polynomials to factor. A Factoring Polynomials of Any Type We begin this section with a checklist that can be used in Factoring Polynomials of any type. When you have finished this section and the problem set that follows, you want to be proficient enough at factoring that the checklist is second nature to you. ©iStockphoto.com/xof711 HOW TO Factor a Polynomial Step 1: If the polynomial has a greatest common factor other than 1, then factor out the greatest common factor. Step 2: If the polynomial has two terms (a binomial), then see if it is the difference of two squares or the sum or difference of two cubes, then factor accordingly. Remember, if it is the sum of two squares, it will not factor. Step 3: If the polynomial has three terms (a trinomial), then either it is a perfect square trinomial, which will factor into the square of a binomial, or it is not a perfect square trinomial, in which case you can try to use the trial and error method developed in Section 5.3. Step 4: If the polynomial has more than three terms, try to factor it by grouping. Step 5: As a final check, see if any of the factors you have written can be factored further. If you have overlooked a common factor, you can catch it here. Remember, some polynomials cannot be factored, in which case they are considered prime. Chapter 5 Factoring Polynomials 420 Here are some examples illustrating how we use the checklist.
eBook - PDFBeginning 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.
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 - PDFIntermediate Algebra
A Guided Approach
- Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 324 CHAPTER 5 Polynomials, Polynomial Functions, and Equations If the greatest common factor of several terms is 1 , the terms are called relatively prime. Determine whether the terms in each set are relatively prime . 131. 14, 45 132. 24, 63, 112 133. 60, 28, 36 134. 55, 49, 78 135. 12 x 2 y , 5 ab 3 , 35 x 2 b 3 136. 18 uv , 25 rs , 12 rsuv SOMETHING TO THINK ABOUT 129. Pick two natural numbers. Divide their product by their greatest common factor. The result is called the least common multiple of the two numbers you picked. Why? 130. The number 6 is called a perfect number , because the sum of all the divisors of 6 is twice 6: 1 1 2 1 3 1 6 5 12 . Verify that 28 is also a perfect number. Section Objectives Vocabulary Factor the difference of two squares. Factor the sum and difference of two cubes. Factor polynomials by grouping involving a difference of squares. The Difference of Two Squares; the Sum and Difference of Two Cubes perfect square difference of squares sum of squares perfect cube sum of cubes difference of cubes 2 3 1 Multiply. 1. 1 a 1 b 21 a 2 b 2 2. 1 5 p 1 q 21 5 p 2 q 2 3. 1 3 m 1 2 n 21 3 m 2 2 n 2 4. 1 2 a 2 1 b 2 21 2 a 2 2 b 2 2 5. 1 a 2 3 21 a 2 1 3 a 1 9 2 6. 1 p 1 2 21 p 2 2 2 p 1 4 2 In this section, we will discuss three special types of factoring problems. These types can be factored using the appropriate factoring formula. Factor the difference of two squares. To factor the difference of two squares, it is helpful to know the first 20 integers that are perfect squares .
No longer available |Learn moreFoundations of Mathematics
Algebra, Geometry, Trigonometry and Calculus
- Philip Brown(Author)
- 2016(Publication Date)
- Mercury Learning and Information(Publisher)
Because any quadratic polynomial has two roots, it can be expressed as a product of two linear factors multiplied by a constant. As a demonstration in table 3.5, we give the factorized form of each of the quadratic polynomials from table 3.3. 68 • Foundations of Mathematics TABLE 3.5. Quadratic polynomials in factorized form Quadratic polynomials Factorized form (i) 2 4 6 2 x x − − 2 1 3 ( )( ) x x + − (ii) − − − x x 2 4 4 − + ( ) x 2 2 (iii) 6 15 2 x x − − + - x x 6 3 2 5 3 (iv) x x 2 11 − − - + - - x x 1 45 2 1 45 2 (v) − − x 2 2 − − + ( 2 )( 2 ) x x i i (vi) + + x x 5 2 2 - + - - x x 1 3i 2 1 3i 2 EXAMPLE 3.5.3. The factorized form for 6 15 2 x x − − (row (iii)) can also be expressed as 6 15 2 3 2 3 5 3 2 3 3 5 2 x x x x x x − − = + − = + − ( )( ). Finding the factors of a polynomial is called factorizing or factoring (we will use the former). It is possible to factorize quadratic polynomials with integer coefficients by inspection if they factorize into a pair of linear factors that also have integer coefficients (e.g., the polynomials in rows (i) and (iii) of the table above). This is a very useful skill, and it is also fun! The trick is to find the correct pairs of factors for the leading coefficient and the constant term. For example, for the polynomial 6 15 2 x x − − (row (iii) in the table), the correct pair of factors of the leading coefficient is 3 and 2 and the correct pair of factors of the constant term is 3 and 5. Finding the correct pairs of factors can be done by trial and error, but it does help to be methodical. We demonstrate this with four sets of examples. EXAMPLE 3.5.4. Factorize, by inspection, each of the following quadratic polynomials.
eBook - PDFIntroductory Algebra
Concepts and Graphs 2E
- Charles P. McKeague(Author)
- 2020(Publication Date)
- XYZ Textbooks(Publisher)
When you have finished this section and the problem set that follows, you want to be proficient enough at factoring that the checklist is second nature to you. Step 1: If the polynomial has a greatest common factor other than 1, then factor out the greatest common factor. Step 2: If the polynomial has two terms (it is a binomial), then see if it is the difference of two squares. Remember, if it is the sum of two squares, it will not factor. Step 3: If the polynomial has three terms (a trinomial), then either it is a perfect square trinomial, which will factor into the square of a binomial, or it is not a perfect square trinomial, in which case you use the trial and error method developed in Section 5.3. Step 4: If the polynomial has more than three terms, try to factor it by grouping. Step 5: As a final check, see if any of the factors you have written can be factored further. If you have overlooked a common factor, you can catch it here. HOW TO: FACTOR A POLYNOMIAL Here are some examples illustrating how we use the checklist. Factor 2x 5 − 8x 3 . SOLUTION First, we check to see if the greatest common factor is other than 1. Since the greatest common factor is 2x 3 , we begin by factoring it out. Once we have done so, we notice that the binomial that remains is the difference of two squares: 2x 5 − 8x 3 = 2x 3 (x 2 − 4) Factor out the greatest common factor, 2x 3 = 2x 3 (x + 2)(x − 2) Factor the difference of two squares Note that the greatest common factor 2x 3 that we factored from each term in the first step of Example 1 remains as part of the answer to the problem; that is because it is one of the factors of the original binomial. Remember, the expression we end up with when factoring must be equal to the expression we start with. We can’t just drop a factor and expect the resulting expression to equal the original expression. EXAMPLE 1 OBJECTIVES A Factor polynomials of any type. 5.6 VIDEOS 330 CHAPTER 5 Factoring Factor 3x 4 − 18x 3 + 27x 2 .
- Rudolf Lidl, Harald Niederreiter(Authors)
- 1994(Publication Date)
- Cambridge University Press(Publisher)
Chapter 4 Factorization of Polynomials Any nonconstant polynomial over a field can be expressed as a product of irreducible polynomials. In the case of finite fields, some reasonably effi- cient algorithms can be devised for the actual calculation of the irreducible factors of a given polynomial of positive degree. The availability of feasible factorization algorithms for polynomials over finite fields is important for coding theory and for the study of linear recurrence relations in finite fields. Beyond the realm of finite fields, there are various computational problems in algebra and number theory that depend in one way or another on the factorization of polynomials over finite fields. We mention the factorization of polynomials over the ring of integers, the determination of the decomposition of rational primes in algebraic number fields, the calculation of the Galois group of an equation over the rationals, and the construction of field extensions. We shall present several algorithms for the factorization of poly- nomials over finite fields. The decision on the choice of algorithm for a specific factorization problem usually depends on whether the underlying finite field is "small" or "large." In Section 1 we describe those algorithms that are better adapted to "small" finite fields and in the next section those that work better for "large" finite fields. Some of these algorithms reduce the problem of Factoring Polynomials to that of finding the roots of certain other polynomials. Therefore, Section 3 is devoted to the discussion of the latter problem from the computational viewpoint. 132 1. Factorization over Small Finite Fields 133 1. FACTORIZATION OVER SMALL FINITE FIELDS Any polynomial/ e ¥ q [x] of positive degree has a canonical factorization in ¥ q [x] by Theorem 1.59. For the discussion of factorization algorithms it will suffice to consider only monic polynomials.
eBook - PDFBeginning and Intermediate Algebra
A Guided Approach
- Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
5.1 Factoring Out the Greatest Common Factor; Factoring by Grouping 307 Section Objectives Vocabulary Identify the greatest common factor of two or more monomials. Factor a polynomial containing a greatest common factor. Factor a polynomial containing a negative greatest common factor. Factor a polynomial containing a binomial greatest common factor. Factor a four-term polynomial using grouping. Factoring Out the Greatest Common Factor; Factoring by Grouping fundamental theorem of arithmetic greatest common factor (GCF) factoring by grouping 2 3 1 4 5 Simplify each expression. 1. 5 1 x 1 3 2 2. 7 1 y 2 8 2 3. x 1 3 x 2 2 2 4. y 1 5 y 1 9 2 5 x 1 15 7 y 2 56 3 x 2 2 2 x 5 y 2 1 9 y 5. 3 1 x 1 y 2 1 a 1 x 1 y 2 6. x 1 y 1 1 2 1 5 1 y 1 1 2 3 x 1 3 y 1 ax 1 ay xy 1 x 1 5 y 1 5 7. 5 1 x 1 1 2 2 y 1 x 1 1 2 8. x 1 x 1 2 2 2 y 1 x 1 2 2 5 x 1 5 2 yx 2 y x 2 1 2 x 2 yx 2 2 y Getting Ready 5.1 In this chapter, we will reverse the operation of multiplication and show how to find the factors of a known product. The process of finding the individual factors of a product is called factoring . We will limit our discussion of Factoring Polynomials to those that factor using only rational numbers. Identify the greatest common factor of two or more monomials. Recall that a natural number greater than 1 whose only factors are 1 and the number itself is called a prime number. The prime numbers less than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47 A natural number is said to be in prime-factored form if it is written as the product of fac-tors that are prime numbers. 1 Copyright 2015 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.
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.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
338 CHAPTER 5 Polynomials and Factoring Notice the third line in the previous example. It consists of all possible products of terms in the first binomial and those of the second binomial. We can generalize this into a rule for multiplying two polynomials. Multiplying polynomials can be accomplished by a method that looks very similar to long multiplication with whole numbers. Multiply (2 x − 3 y ) and (3 x 2 − xy + 4 y 2 ) vertically. SOLUTION 3 x 2 − xy + 4 y 2 2 x − 3 y 6 x 3 − 2 x 2 y + 8 xy 2 Multiply (3 x 2 − xy + 4 y 2 ) by 2 x − 9 x 2 y + 3 xy 2 − 12 y 3 Multiply (3 x 2 − xy + 4 y 2 ) by − 3 y 6 x 3 − 11 x 2 y + 11 xy 2 − 12 y 3 Add similar terms Multiplying Binomials—The FOIL Method Consider the product of (2 x − 5) and (3 x − 2). Distributing (3 x − 2) over 2 x and − 5, we have (2 x − 5)(3 x − 2) = (2 x )(3 x − 2) + ( − 5)(3 x − 2) = (2 x )(3 x ) + (2 x )( − 2) + ( − 5)(3 x ) + ( − 5)( − 2) = 6 x 2 − 4 x − 15 x + 10 = 6 x 2 − 19 x + 10 Looking closely at the second and third lines, we notice the following: 1. 6 x 2 comes from multiplying the first terms in each binomial. (2 x − 5)(3 x − 2) 2 x (3 x ) = 6 x 2 First terms 2. − 4 x comes from multiplying the outside terms in the product. (2 x − 5)(3 x − 2) 2 x ( − 2) = − 4 x Outside terms 3. − 15 x comes from multiplying the inside terms in the product. (2 x − 5)(3 x − 2) − 5(3 x ) = − 15 x Inside terms 4. 10 comes from multiplying the last two terms in the product. (2 x − 5)(3 x − 2) − 5( − 2) = 10 Last terms To multiply two polynomials, multiply each term in the first polynomial by every term in the second polynomial. RULE Multiplication of Polynomials EXAMPLE 3 Note The vertical method of multiplying polynomials does not directly show the use of the distributive property. It is, however, very useful since it always gives the correct result and is easy to remember. Note The FOIL method does not show the properties used in multiplying two binomials.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2017(Publication Date)
- XYZ Textbooks(Publisher)
6.5 Factoring: A General Review 419 Factor: 2 ab 5 + 8 ab 4 + 2 ab 3 . SOLUTION The greatest common factor is 2 ab 3 . We begin by factoring it from each term. After that we find the trinomial that remains cannot be factored further: 2 ab 5 + 8 ab 4 + 2 ab 3 = 2 ab 3 ( b 2 + 4 b + 1) Factor: xy + 8 x + 3 y + 24. SOLUTION Since our polynomial has four terms, we try factoring by grouping: xy + 8 x + 3 y + 24 = x ( y + 8) + 3( y + 8) = ( y + 8)( x + 3) After reading through the preceding section, respond in your own words and in complete sentences. A. What is the first step in factoring any polynomial? B. If a polynomial has four terms, what method of factoring should you try? C. If a polynomial has two terms, what method of factoring should you try? D. What is the last step in factoring any polynomial? Getting Ready for Class EXAMPLE 6 EXAMPLE 7 Problem Set 6.5 420 Factor each of the following polynomials completely; that is, once you are fin-ished factoring, none of the factors you obtain should be factorable any further. Also, note that the even-numbered problems are not necessarily similar to the odd-numbered problems that precede them in this problem set. 1. x 2 − 81 2. x 2 − 18 x + 81 3. x 2 + 2 x − 15 4. 15 x 2 + 11 x − 6 5. x 2 + 6 x + 9 6. 12 x 2 − 11 x + 2 7. y 2 − 10 y + 25 8. 21 y 2 − 25 y − 4 9. 2 a 3 b + 6 a 2 b + 2 ab 10. 6 a 2 − ab − 15 b 2 11. x 2 + x + 1 12. 2 x 2 − 4 x + 2 13. 12 a 2 − 75 14. 16 a 3 − 250 15. 9 x 2 − 12 xy + 4 y 2 16. x 3 − x 2 17. 4 x 3 + 16 xy 2 18. 16 x 2 + 49 y 2 19. 2 y 3 + 20 y 2 + 50 y 20. 3 y 2 − 9 y − 30 21. a 6 + 4 a 4 b 2 22. 5 a 2 − 45 b 2 23. xy + 3 x + 4 y + 12 24. xy + 7 x + 6 y + 42 25. x 4 − 16 26. x 4 − 81 27. xy − 5 x + 2 y − 10 28. xy − 7 x + 3 y − 21 29. 5 a 2 + 10 ab + 5 b 2 30. 3 a 3 b 2 + 15 a 2 b 2 + 3 ab 2 31. 64 + x 3 32. 49 + x 2 33. 3 x 2 + 15 xy + 18 y 2 34. 3 x 2 + 27 xy + 54 y 2 35. 2 x 2 + 15 x − 38 36. 2 x 2 + 7 x − 85 37. 100 x 2 − 300 x + 200 38. 100 x 2 − 400 x + 300 39. x 2 − 64 40.
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