Factoring Polynomials

Save this PDF as:

Size: px
Start display at page: Transcription

1 Factoring Polynomials Hoste, Miller, Murieka September 12, Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations below, where we multiply the expressions on the left to produce the expressions on the right: x(x + 2) = x 2 + 2x (w 7)(w + 7) = w (z 3)(z + 2)(z 2 + 5) = 2z 4 2z 3 2z 2 10z 60 In this section, we discuss how to undo this process. There are times when the expression on the right is much simpler to work with. However, the left side can provide valuable information as well. We want to be able to take an expression in either form, and convert it to the other. In this section, we discuss how to undo the process of multiplication. This is called factoring. We will see later that factoring is often used to answer the very important question: For what values of a variable is an expression equal to zero? So we see that there are now two points of view from which to look at an equation like: (r 6)(r + 8) = r 2 + 2r 48 These are: 1. The product of (r 6) and (r + 8) may be expressed as r 2 + 2r The factors of the expression r 2 + 2r 48 are (r 6) and (r + 8). 1

2 In the previous section, we saw that multiplication of algebraic terms is very straightforward, in fact what we might call cook book. Factoring isn t quite as simple. There are several rules and formulas to recall (or look up), and it is crucial to know when to use these. In this section, we will discuss these rules, and how to use them, providing several examples of each. Below we will list several rules used to factor algebraic expressions. Most of these are factoring instructions, based on the form of the expression needing to be factored. In this section, we will show how to factor only polynomial expressions. Later, when we encounter more complicated types of expressions, we can easily show how the ideas presented here can be generalized. I Factoring an Expression Using Simple Formulas These first simple formulas involve the sums and differences of the same power of two terms. Their simplicity is due to the fact that the the power is low, either 2 or 3. Later we will generalize a few of these, using higher powered terms. Method 1 The Difference of Squares x 2 a 2 = (x a)(x + a) Example 1 x 2 64 = (x 8)(x + 8) v = (v 1 14 )(v ) w 2 5 = (w 5)(w + 5) h 2 2b 2 = (h 2b)(h + 2b) z 2 r 6 = (z r 3 )(z + r 3 ) n = (n )(n ) Method 2 The Difference of Cubes x 3 a 3 = (x a)(x 2 + ax + a 2 ) 2

3 Example 2 x 3 64 = (x 4)(x 2 + 4x + 16) v = (v 11)(v v + 121) r 3 1 = (r 1)(r 2 + r + 1) t 3 π 3 = (t π)(t 2 + πt + π 2 ) n 3 4 = (n )(n n ) b = (b 1 9 )(b2 + b ) Method 3 The Sum of Cubes x 3 + a 3 = (x + a)(x 2 ax + a 2 ) r = (r + 1)(r 2 r + 1) Example 3 x = (x + 4)(x 2 4x + 16) t π 3 = (t + π 1 )(t 2 π 1 t + π 2 ) 27L 3 + (2π) 3 = (3L + 2π)(9L 2 6Lπ + 4π 2 ) z = (z + 12)(z 2 12z + 144) y = (y )(y y ) Method 4 Factoring out the Greatest Common Divisor A common divisor, found in each term of a polynomial, should be factored and placed in front of the expression, which must now be placed in parentheses. This common divisor might be either 1) a constant, 2) the lowest power of the variable, or 3) a product of both. 3

4 4x 4 28x x 16 = 4(x 4 7x 2 + 5x 4) Example 4 7z 2 70 = 7(z 2 10) = 7(z 10)(z + 10) x 8 + 4x 4 + 7x 2 = x 2 (x 6 + 4x 2 + 7) x 5 30x 3 = x 3 (x 2 30) = x 3 (x 30)(x + 30) 18x x 9 48x 3 9x 5 = 3(6x 4 + 8x 9 16x 3 3x 5 ) = 3x 3 (6x + 8x x 2 ) 5x x x 3 5 = x 3 5 (5x 3 2x ) At this point, before introducing Method 5, we introduce an important fact, or general principle which will help guide us in factoring. We will not discuss why this is true, but simply present it as a guide. It might seem a bit dry or theoretic, and yet at the same time, a very powerful and wide reaching statement about factoring! Important Fact 1 The Fundamental Theorem of Algebra Any polynomial (in a single variable) can be factored to the point where all of it s factors are either linear or quadratic This is very interesting to mathematicians! Throughout the study of mathematics, you will come across these so called Fundamental Theorems. As the name suggests, these are the most important facts in any area of mathematics. We will see more of these! In this case, Important Fact 1 is one version of the Fundamental Theorem of Algebra. Many of the Fundamental Theorems have two or three slightly different versions, in order to be specifically applied to lots of different problems! In theory, the fundamental theorem of algebra promises that any polynomial has some set of factors, all of which are either linear or quadratic. However, in practice, it is not always easy, or even possible, to factor a polynomial this completely, using the methods in this chapter. Try factoring the polynomial: x 4 + 2x 3 13x 2 14x + 24, using the rules Example 5 we have presented above. None of them seem to apply. There is, at this point, no simple way to determine the factors of this polynomial. Are they all linear? Is one or more factor quadratic? Important Fact 1 tells us that 4

5 there are factors of this polynomial which are either linear or quadratic. Even though we can t find these factors, once we know what they are, it is easy to verify that they are correct. In particular, we can check that: (x + 4)(x 3)(x + 2)(x 1) = x 4 + 2x 3 13x 2 14x + 24 To make things even more complicated, the coefficients of the factors may not even be rational numbers, even if the coefficients of the original polynomial are! We have already seen several examples of this above, such as: w 2 5 = (w 5)(w + 5) The truth is, that at this point, we can only give a few rules which can be applied to some very special examples. As a general rule, the lower the degree of the polynomial, the better our chances are of factoring it. There is good news however: First, in actual practice, many of the problems seen in the real world assume the form of simple polynomials which can be easily factored by using the rules we present here. Secondly, in future sections, we will show very powerful ways to find the factors of a wide variety of polynomials! II Method 5: Factoring a Quadratic by Trial and Error Any quadratic expression can be written as Ax 2 + Bx + C for some real numbers, A, B, and C, and variable x. Using Method 4, if possible, we may always assume that A, B, and C are as simple as possible, and that A is positive. For example: 3x 2 + 2x 5 = 1(3x 2 2x + 5) 24x 2 + 8x 4 = 4(6x 2 + 2x 1) Using Method 4 is not absolutely essential, but does simplify what follows. One might think that such a simple expression should easily factor, using some very simple rules, but this is not always true. However, if a 5

6 quadratic polynomial can be factored, the result is always a product of two linear factors. This means that if a quadratic expression is factorable at all, it must have the form: Ax 2 + Bx + C = (dx + e)(gx + h) for some real numbers d, e, g, h. Our job, then, is to determine the relationship between the constants A, B, C and d, e, g, h. That will be the basis of our trial and error method. Expanding (dx + e)(gx + h), we obtain: (dx + e)(gx + h) = dgx 2 + dhx + egx + eh = dgx 2 + (dh + eg)x + eh = Ax 2 + Bx + C This single relationship gives us 3 equations: A = dg B = (dh + eg) C = eh These are easily verified in the following examples: Ax 2 + Bx + C = (dx + e) (gx + h) Example 6 x 2 + 7x + 10 = (x + 2) (x + 5) x 2 9x + 20 = (x 4) (x 5) x 2 + 3x 18 = (x + 6) (x 3) x 2 5x 14 = (x + 2) (x 7) 8x 2 10x 3 = (2x 3) (4x + 1) 6x 2 10x 3 = 3(x 3) (2x + 5) 6

7 Notice how the signs in the factors affect the signs in the product. We can summarize the behavior by listing the four possibilities: Ax 2 + Bx + C = (dx + e)(gx + h) Ax 2 Bx + C = (dx e)(gx h) Ax 2 + Bx C = (dx + e)(gx h) Ax 2 Bx C = (dx + e)(gx h) Let s see how these properties can help factor a quadratic polynomial. We use the four step trial and error procedure below: 1. Noting the signs of the quadratic coefficients, set up the parentheses for the two factors, and insert the correct signs: Example: 4x 2 7x + 3 = ( x )( x ) 2. Recalling the rule A = dg, we guess the two x-coefficients, knowing that their product must equal the leading coefficient of the quadratic polynomial. There will usually be more than one such set which looks reasonable. Example: 4x 2 7x + 3 = (2x )(2x ) or = (4x )(1x ) 3. Recalling the rule C = eh, we guess the two constant coefficients, knowing that their product must equal the constant coefficient of the quadratic polynomial. There will usually be more than one such set which looks reasonable. Example: 4x 2 7x + 3 = (2x 1)(2x 3) or = (2x 3)(2x 1) or = (4x 1)(1x 3) or = (4x 3)(1x 1) 7

8 4. For each such trial factorization, multiply the factors to determine if the product is, in fact, the quadratic polynomial you started with. If so, you have found your factors. If not, keep looking until you have checked all possibilities Example: (4x 3)(1x 1) = 4x 2 3x 4x + 3 = 4x 2 7x + 3 Consider the polynomial 5x x 6. One way to perform the 4 steps above quickly is to write the potential factorization out similar to: { 5 5x x 6 = ( 1 } x )( { 1 5 } x ) Example 7 The above is really just a shorthand notation for steps 1, 2, and 3. To perform Step 4, we must multiply every combination of numbers from {5,1} with numbers from {6, 3, 2, 1}. There will be 8 possible products: 30, 15, 10, 5, 6, 3, 2, 1. If one of these 8 products, minus another of these is equal to 29, the middle term of the polynomial, then we have found it s factors! In this case, since 30 1 = 29, (or specifically, = 29), we know that our factorization is: { 5 5x 2 +29x 6 = ( 1 } x { 1 )( 5 } x ) = (x+6)(5x 1) which, of course, we will verify by expanding (x + 6)(5x 1) and seeing that we obtain 5x x 6. A few final words about Method 5. First, we emphasize one more time, that some quadratic polynomials can not be factored. Important Fact 1 says 8

9 that every polynomial can be factored down to the product of linear and quadratic terms, but does not tell us any more than this. Secondly, notice that in steps 2) and 3) above, we are really looking for constant divisors of the coefficients of the leading and constant terms of the quadratic polynomial. Finding these gives us the list of possible factors! Finally, the astute reader has possibly already wondered if steps 1), 2) and 3) above are interchangeable. In fact they are. We can look at the linear coefficients first, or the constant terms first, or place the + or signs early or later... whichever makes the most sense to the individual. III Some advanced cases Below we show some examples which at first seem more complicated. Yet we simply use the same rules above... but a bit more creatively. In particular, we often combine some of the rules above. 9

10 Example 8 v = (v 2 25)(v ) = (v 5)(v + 5)(v ) Method 1 used twice y = (y 4 16)(y ) = (y 2 4)(y 2 + 4)(y ) = (y 2)(y + 2)(y 2 + 4)(y ) Method 1 used 3 times (3x 5) 2 36 = ((3x 5) 6)((3x 5) + 6) = (3x 11)(3x + 1) Method 1 applied using the variable (3x 5) v 4 12v = (v 2 7)(v 2 5) = (v 7)(v + 7)(v 5)(v + 5) Method 5 applied using the variable v 2, then Method 1 applied twice using the variable v w 6 117w = (w 3 125)(w 3 + 8) = (w 5)(w 2 + 5w + 25)(w + 2)(w 2 2w + 4) Method 5 applied using the variable w 3, then Methods 2 and 3 applied to the cubic factors r 7 b 3 rb 5 = rb 3 (r 6 b 2 ) = rb 3 (r 3 b)(r 3 + b) = rb 3 (r b 1 3 )(r 2 + rb b 2 3 )(r + b 1 3 )(r 2 rb b 2 3 ) Method 4, then Method 1, then Methods 2 and 3 40v = 5(8v 3 + 1) = 5(2v + 1)(4v 2 2v + 1) Method 4, then Method 3 63v 7 28v 3 = 7v 3 (9v 4 4) = 7v 3 (3v 2 2)(3v 2 + 2) = 7v 3 ( 3v 2)( 3v + 2)(3v 2 + 2) Method 4, then Method 1 twice 2 m4 z 3 11 m3 z 2 6 m2 = m2 (2m 2 11mz 6z 2 ) = m2 (m 6z)(2m + z) z z 3 z 3 Method 4, then Method 5 10

11 IV A Few Higher Degree Factoring Formulas Look again at Methods 1, 2 and 3. We can give higher order analogs of these formulas. We leave it to the curious or doubting reader to verify that these formulas are correct. Hint: Multiply the factors! Method 5 The Difference of Positive Integer Powers If n is a positive integer, then: x n a n = (x a)(x n 1 +ax n 2 +a 2 x n 3 + +a n 3 x 2 +a n 2 x+a n 1 ) Method 6 The Sum of Odd Integer Powers If n is a positive odd integer, then: x n +a n = (x+a)(x n 1 ax n 2 +a 2 x n 3 +a n 3 x 2 a n 2 x+a n 1 ) x 6 64 = (x 2)(x 5 + 2x 4 + 4x 3 + 8x2 + 16x + 32) Example 9 x = (x )(x x x x ) x n 1 = (x 1)(x n 1 + x n 2 + x n x 2 + x + 1) for any integer n > 1 We just verified in the previous example that we can factor x 6 64 to get: Example 10 x 6 64 = (x 2)(x 5 + 2x 4 + 4x 3 + 8x2 + 16x + 32). Yet, using Method 1, then 2, we get: x 6 64 = (x 3 8)(x 3 + 8) = (x 2)(x 2 + 2x + 4)(x + 2)(x 2 2x + 4) On the other hand, using Method 2, then 1, we get: x 6 64 = (x 2 4)(x 4 + 4x ) = (x 2)(x + 2)(x 4 + 4x ) 11

12 Comparing all of these factors, we see that : (x 5 + 2x 4 + 4x 3 + 8x2 + 16x + 32) = (x 2 + 2x + 4)(x + 2)(x 2 2x + 4) which further tells us that: = (x + 2)(x 4 + 4x ) (x 4 + 4x ) = (x 2 + 2x + 4)(x 2 2x + 4). The moral to this story: There is often more than one way to factor a polynomial and get different factors. However, there is one and only one set of prime factors for any given polynomial, where a prime factor is one that cannot be factored any further. Until we reach a factorization using only prime factors, we often see a vastly different set of factors, depending on which method we use. To see this more clearly, consider an analogous example of factoring an integer, for instance, the number 56. If asked to factor 56, we might say 56 = 8 7, and someone else might say 56 = Both of these are true statements and the factors in each case are different. But the astute observer should see that some of the factors can be factored further. Eventually, if we factor as much as possible, we are left with a set of factors which are all prime numbers: 56 = Exercises for Section 0.1 In problems 1 through 5, factor the polynomials as completely as possible. 1. (a) v (b) w 4 49 (c) s 3 8 (d) x (e) r (f) y 3 50 (g) 3v 3 81 (h) 5w 5 320w 2 (i) 11v 2 22v + 11 (j) w 4 7w 2 30 (k) v v (l) L 100 4L 99 21L 98 (m) r 25 6r r 15 (n) (y 3) 2 +5(y 3) 14(y 3) 12

13 2. (a) x 2 7x + 12 (b) x 2 3x 10 (c) x 2 + 4x 12 (d) x 2 + 5x 24 (e) x 2 + 9x + 14 (f) x x + 16 (g) x x + 24 (h) 2x 2 + x 1 (i) 8x 2 6x + 1 (j) 12x 2 x 1 (k) 4x x + 6 (l) 6x 2 25x + 4 (m) x x 1296 (n) x 2 6x (a) x 4 7x (b) x 18 3x 9 10 (c) ( 1 x )2 + 4 x 12 (d) x 4 + 5x 2 24 (e) (5x + 3) 6 + 9(5x + 3) (f) x x x 3 4. (a) u 7 3 4u 1 3 (b) v 19 7 v 2 7 (c) w w 19 2 (d) r 5 8r 3 5. (a) x (b) x (c) x 8 1 (d) x 9 1 (e) x (f) x Identify as many factors of the polynomial x as possible. Hint: Use as many of the factoring methods as you can. 7. Consider the polynomial x a) Verify that this quartic polynomial has the following two quadratic factors (x 2 2x + 2)(x 2 + 2x + 2) b) What are the factors for the quartic polynomial x 4 + C for any positive number C? Hint: Assume the form x 4 + C = (x 2 Rx + S)(x 2 + Rx + S), multiply out the factors, and find R and S in terms of C. c) Factor the polynomial x d) Factor the polynomial x Compute the value of , without actually performing the sum, but using a clever formula. 13

14 9. Compute the value of actually performing the sum, but using a clever formula , without 14

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

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

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

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

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

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4) ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

3.6. The factor theorem 3.6. The factor theorem Example 1. At the right we have drawn the graph of the polynomial y = x 4 9x 3 + 8x 36x + 16. Your problem is to write the polynomial in factored form. Does the geometry of the

Section 6.1 Factoring Expressions Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what

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

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

Vieta s Formulas and the Identity Theorem Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

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,...}

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m) Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

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

1 Lecture: Integration of rational functions by decomposition Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

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

The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11. 9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

MATH 90 CHAPTER 6 Name:. MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a

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

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 Any Any Any natural number that that that greater greater than than than 1 1can can 1 be can be be factored into into into a a a product of of of prime prime numbers. For For For

Solving Cubic Polynomials Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial

Partial Fractions. (x 1)(x 2 + 1) Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

FACTORING QUADRATICS 8.1.1 and 8.1.2 FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.

( ) 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

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned 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 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

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

Integer roots of quadratic and cubic polynomials with integer coefficients Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

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,

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.

Factoring Polynomials Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

minimal polyonomial Example Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

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.

3.6 The Real Zeros of a Polynomial Function SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,

Factoring A Quadratic Polynomial 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

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

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.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

Determinants can be used to solve a linear system of equations using Cramer s Rule. 2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

AIP Factoring Practice/Help The following pages include many problems to practice factoring skills. There are also several activities with examples to help you with factoring if you feel like you are not proficient with it. There

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

Integrals of Rational Functions Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t

Solving Quadratic Equations by Factoring 4.7 Solving Quadratic Equations by Factoring 4.7 OBJECTIVE 1. Solve quadratic equations by factoring The factoring techniques you have learned provide us with tools for solving equations that can be written

is identically equal to x 2 +3x +2 Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3

What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same

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

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

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

EAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

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.

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

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/

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

3 1. Note that all cubes solve it; therefore, there are no more Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if

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

Winter Camp 2011 Polynomials Alexander Remorov. Polynomials. Alexander Remorov alexanderrem@gmail.com Polynomials Alexander Remorov alexanderrem@gmail.com Warm-up Problem 1: Let f(x) be a quadratic polynomial. Prove that there exist quadratic polynomials g(x) and h(x) such that f(x)f(x + 1) = g(h(x)).

1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x). .7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational

APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic

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,

3.2 The Factor Theorem and The Remainder Theorem 3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial

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

Tim Kerins. Leaving Certificate Honours Maths - Algebra. Tim Kerins. the date Leaving Certificate Honours Maths - Algebra the date Chapter 1 Algebra This is an important portion of the course. As well as generally accounting for 2 3 questions in examination it is the basis for many

HIBBING COMMUNITY COLLEGE COURSE OUTLINE HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam, Factorising quadratics An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to

it is easy to see that α = a 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore 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

CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

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

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

is the degree of the polynomial and is the leading coefficient. Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...

Linear and quadratic Taylor polynomials for functions of several variables. ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is

Factoring Polynomials Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

Copyrighted Material. Chapter 1 DEGREE OF A CURVE Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

6.1 Add & Subtract Polynomial Expression & Functions 6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

UNCORRECTED PAGE PROOFS number and and algebra TopIC 17 Polynomials 17.1 Overview Why learn this? Just as number is learned in stages, so too are graphs. You have been building your knowledge of graphs and functions over time.

Unit 3: Day 2: Factoring Polynomial Expressions Unit 3: Day : Factoring Polynomial Expressions Minds On: 0 Action: 45 Consolidate:10 Total =75 min Learning Goals: Extend knowledge of factoring to factor cubic and quartic expressions that can be factored

Polynomial Expressions and Equations Polynomial Expressions and Equations This is a really close-up picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F. FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods

Solving Rational Equations Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply, Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity: 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