6.6 Factoring Strategy
|
|
- Jeffry Reynolds
- 4 years ago
- Views:
Transcription
1 456 CHAPTER 6. FACTORING 6.6 Factoring Strategy When you are concentrating on factoring problems of a single type, after doing a few you tend to get into a rhythm, and the remainder of the exercises, because they are similar, seem to flow. However, when you encounter a mixture of factoring problems of different types, progress is harder. The goal of this section is to set up a strategy to follow when attacking a general factoring problem. If it hasn t already been done, it is helpful to arrange the terms of the given polynomial in some sort of order (descending or ascending). Then you want to apply the following guidelines. Factoring Strategy. These steps should be followed in the order that they appear. 1. Factor out the greatest common factor (GCF). 2. Look for a special form. a) If you have two perfect squares separated by a minus sign, use the difference of squares pattern to factor: a 2 b 2 = (a+b)(a b) b) If you have a trinomial whose first and last terms are perfect squares, you should suspect that you have a perfect square trinomial. Take the square roots of the first and last terms and factor as follows. a 2 +2ab+b 2 = (a+b) 2 Be sure to check that the middle term is correct. 3. If you have a trinomial of the form ax 2 + bx+c, use the ac-method to factor. 4. If you have a four-term expression, try to factor by grouping. Once you ve applied the above strategy to the given polynomial, it is quite possible that one of your resulting factors will factor further. Thus, we have the following rule. Factor completely. The factoring process is not complete until none of your remaining factors can be factored further. This is the meaning of the phrase, factor completely.
2 6.6. FACTORING STRATEGY 457 Finally, a very good word of advice. Check your factoring by multiplying. Once you ve factored the given polynomial completely, it is a very good practice to check your result. If you multiply to find the product of your factors, and get the original given polynomial as a result, then you know your factorization is correct. It s a bit more work to check your factorization, but it s worth the effort. It helps to eliminate errors and also helps to build a better understanding of the factoring process. Remember, factoring is unmultiplying, so the more you multiply, the better you get at factoring. Let s see what can happen when you don t check your factorization! Warning! The following solution is incorrect! Factor: 2x 4 +8x 2. Solution: Factor out the GCF. 2x 4 +8x 2 = 2x 2 (x 2 +4) = 2x 2 (x+2) 2 Note that this student did not bother to check his factorization. Let s do that for him now. Check: Multiply to check. Remember, when squaring a binomial, there is a middle term. 2x 2 (x+2) 2 = 2x 2 (x 2 +4x+4) = 2x 4 +8x 3 +8x 2 This is not the same as the original polynomial 2x 4 + 8x 2, so the student s factorization is incorrect. Had the student performed this check, he might have caught his error, provided of course, that he multiplies correctly during the check. The correct factorization follows. 2x 4 +8x 2 = 2x 2 (x 2 +4) The sum of squares does not factor, so we are finished. Check: Multiply to check. 2x 2 (x 2 +4) = 2x 4 +8x 2 This is the same as the original polynomial 2x 4 +8x 2, so this factorization is correct.
3 458 CHAPTER 6. FACTORING 4x 7 +64x 3 EXAMPLE 1. 3x 6 +3x 2 Solution: The first rule of factoring is Factor out the GCF. The GCF of 3x 6 and 3x 2 is 3x 2, so we could factor out 3x 2. 3x 6 +3x 2 = 3x 2 ( x 4 +1) This is perfectly valid, but we don t like the fact that the second factor starts with x 4. Let s factor out 3x 2 instead. 3x 6 +3x 2 = 3x 2 (x 4 1) The second factor is the difference of two squares. Take the square roots, separating one pair with a plus sign, one pair with a minus sign. = 3x 2 (x 2 +1)(x 2 1) The sum of squares does not factor. But the last factor is the difference of two squares. Take the square roots, separating one pair with a plus sign, one pair with a minus sign. Check: Multiply to check the result. = 3x 2 (x 2 +1)(x+1)(x 1) 3x 2 (x 2 +1)(x+1)(x 1) = 3x 2 (x 2 +1)(x 2 1) = 3x 2 (x 4 1) = 3x 6 +3x 2 Answer: 4x 3 (x 2 +4)(x+2)(x 2) The factorization checks. 3a 2 b 4 +12a 4 b 2 12a 3 b 3 EXAMPLE 2. x 3 y +9xy 3 +6x 2 y 2 Solution: The first rule of factoring is Factor out the GCF. The GCF of x 3 y, 9xy 3, and 6x 2 y 2 is xy, so we factor out xy. x 3 y +9xy 3 +6x 2 y 2 = xy(x 2 +9y 2 +6xy) Let s order that second factor in descending powers of x. = xy(x 2 +6xy +9y 2 )
4 6.6. FACTORING STRATEGY 459 The first and last terms of the trinomial factor are perfect squares. We suspect we have a perfect square trinomial, so we take the square roots of the first and last terms, check the middle term, and write: Thus, x 3 y +9xy 3 +6x 2 y 2 = xy(x+3y) 2. Check: Multiply to check the result. = xy(x+3y) 2 xy(x+3y) 2 = xy(x 2 +6xy +9y 2 ) = x 3 y +6x 2 y 2 +9xy 3 Except for the order, this result is the same as the given polynomial. The factorization checks. Answer: 3a 2 b 2 (2a b) 2 EXAMPLE 3. 2x 3 48x+20x 2 Solution: In the last example, we recognized a need to rearrange our terms after we pulled out the GCF. This time, let s arrange our terms in descending powers of x right away. 27x 3 3x 4 60x 2 Now, let s factor out the GCF. 2x 3 48x+20x 2 = 2x 3 +20x 2 48x = 2x(x 2 +10x 24) The last term of the trinomial factor is not a perfect square. Let s move to the ac-method to factor. The integer pair 2,12 has a product equal to ac = 24 and a sum equal to b = 10. Because the coefficient of x 2 is one, this is a drop in place situation. We drop our pair in place and write: Thus, 2x 3 48x+20x 2 = 2x(x 2)(x+12). = 2x(x 2)(x+12) Check: Multiply to check the result. We use the FOIL method shortcut and mental calculations to speed things up. 2x(x 2)(x+12) = 2x(x 2 +10x 24) = 2x 3 +20x 2 48x Except for the order, this result is the same as the given polynomial. The factorization checks. Answer: 3x 2 (x 4)(x 5)
5 460 CHAPTER 6. FACTORING 8x 2 +14xy 15y 2 EXAMPLE 4. 2a 2 13ab 24b 2 Solution: There is no common factor we can factor out. We have a trinomial, butthefirstandlasttermsarenotperfectsquares,solet sapplytheac-method. Ignoring the variables for a moment, we need an integer pair whose product is ac = 48 and whose sum is 13. The integer pair 3, 16 comes to mind (if nothing comes to mind, start listing integer pairs). Break up the middle term into a sum of like terms using the integer pair 3, 16, then factor by grouping. 2a 2 13ab 24b 2 = 2a 2 +3ab 16ab 24b 2 = a(2a+3b) 8b(2a+3b) = (a 8b)(2a+3b) Thus, 2a 2 13ab 24b 2 = (a 8b)(2a+3b). Check: Multiply to check the result. We use the FOIL method shortcut and mental calculations to speed things up. (a 8b)(2a+3b) = 2a 2 13ab 24b 2 Answer: (2x+5y)(4x 3y) This result is the same as the given polynomial. The factorization checks. 36x 3 +60x 2 +9x EXAMPLE 5. 30x 4 +38x 3 20x 2 Solution: The first step is to factor out the GCF, which in this case is 2x 2. 30x 4 +38x 3 20x 2 = 2x 2 (15x 2 +19x 10) The first and last terms of the trinomial factor are not perfect squares, so let s moveagaintotheac-method. Comparing15x 2 +19x 10with ax 2 +bx+c,note that ac = (15)( 10) = 150. We need an integer pair whose product is 150 and whose sum is 19. The integer pair 6 and 25 satisfies these requirements. Because a 1, this is not a drop in place situation, so we need to break up the middle term as a sum of like terms using the pair 6 and 25. = 2x 2 (15x 2 6x+25x 10) Factor by grouping. Factor 3x out of the first two terms and 5 out of the third and fourth terms. = 2x 2 (3x(5x 2)+5(5x 2))
6 6.6. FACTORING STRATEGY 461 Finally, factor out the common factor 5x 2. = 2x 2 (3x+5)(5x 2) Thus, 30x 4 +38x 3 20x 2 = 2x 2 (3x+5)(5x 2). Check: Multiply to check the result. Use the FOIL method to first multiply the binomials. Distribute the 2x 2. 2x 2 (3x+5)(5x 2) = 2x 2 (15x 2 +19x 10) = 30x 4 +38x 3 20x 2 This result is the same as the given polynomial. The factorization checks. Answer: 3x(6x + 1)(2x + 3) EXAMPLE 6. 8x 5 +10x 4 72x 3 90x 2 Solution: Each of the terms is divisible by 3x 3. Factor out 3x 3. 15x 6 33x 5 240x x 3 = 3x 3[ 5x 3 11x 2 80x+176 ] 15x 6 33x 5 240x x 3 The second factor is a four-term expression. Factor by grouping. = 3x 3[ x 2 (5x 11) 16(5x 11) ] = 3x 3 (x 2 16)(5x 11) The factorx 2 16isadifferenceoftwosquares. Takethe squareroots, separate one pair with a plus, one pair with a minus. = 3x 3 (x+4)(x 4)(5x 11) Thus, 15x 6 33x 5 240x x 3 = 3x 3 (x+4)(x 4)(5x 11). Check: Multiply to check the result. 3x 3 (x+4)(x 4)(5x 11) = 3x 3 (x 2 16)(5x 11) = 3x 3 (5x 3 11x 2 80x+176 = 15x 6 33x 5 240x x 3 This result is the same as the given polynomial. The factorization checks. Answer: 2x 2 (x 3)(x+3)(4x+5)
7 462 CHAPTER 6. FACTORING Using the Calculator to Assist the ac-method When using the ac-method to factor ax 2 +bx+c and ac is a very large number, then it can be difficult to find a pair whose product is ac and whose sum in b. For example, consider the trinomial: 12y 2 11y 36 Weneedanintegerpairwhoseproductisac = 432andwhosesumis b = 11. We begin listing integer pair possibilities, but the process quickly becomes daunting. 1, 432 2, 216 Note that the numbers in the second column are found by dividing ac = 432 by the number in the first column. We ll now use this fact and the TABLE feature on our calculator to pursue the desired integer pair. 1. Enter the expression -432/X into Y1 in the Y= menu (see the first image in Figure 6.30). 2. Above the WINDOW button you ll see TBLSET. Use the 2nd key, then press the WINDOW button to access the menu shown in the second image of Figure Set TblStart=1, Tbl=1, then highlight AUTO for both the independent and dependent variables. 3. Above the GRAPH button you ll see TABLE. Use the 2nd key, then press the GRAPH button to access the table shown in the third image in Figure Use the up- and down-arrow keys to scroll through the contents of the table. Note that you can ignore most of the pairs, because they are not both integers. Pay attention only when they are both integers. In this case, remember that you are searching for a pair whose sum is b = 11. Note that the pair 16, 27 shown in the third image of Figure 6.30 is the pair we seek. Figure 6.30: Using the TABLE feature to assist the ac-method.
8 6.6. FACTORING STRATEGY 463 Now we can break the middle term of 12y 2 11y 36 into a sum of like terms using the ordered pair 16, 27, then factor by grouping. 12y 2 11y 36 = 12y 2 +16y 27y 36 = 4y(3y+4) 9(3y+4) = (4y 9)(3y +4) Check: Use the FOIL method shortcut and mental calculations to multiply. The factorization checks. (4y 9)(3y +4) = 12y 2 11y 36
9 464 CHAPTER 6. FACTORING Exercises In Exercises 1-12, factor each of the given polynomials completely y 4 z 2 144y 2 z s 4 t 4 242s 2 t x 7 z 5 363x 5 z r 5 s 2 80r 3 s u 7 162u x 4 320x v v a 9 48a x 6 300x y 5 18y u 7 w 3 2u 3 w y 8 z 4 3y 4 z 8 In Exercises 13-24, factor each of the given polynomials completely a 6 210a a v 7 560v v a 5 b a 4 b a 3 b u 6 v u 5 v u 4 v b 5 +4b 4 +2b v 6 +30v 5 +75v z 4 4z 3 +2z u 6 40u u x x x b 4 +84b 3 +18b b 4 c 5 240b 3 c b 2 c a 5 c 4 180a 4 c 5 +50a 3 c 6 In Exercises 25-36, factor each of the given polynomials completely a 5 +5a 4 210a y 5 9y 4 12y y 6 39y y y 7 27y 6 +42y z 4 +12z 3 135z a 4 40a 3 45a a 6 +64a a x 4 +64x x z 4 +33z 3 +84z a 6 +65a a z 7 75z z y 4 27y 3 +24y 2
10 6.6. FACTORING STRATEGY 465 In Exercises 37-48, factor each of the given polynomials completely b 3 22b 2 +30b 38. 4b 6 22b 5 +30b u 4 w 5 3u 3 w 6 20u 2 w x 5 z 2 +9x 4 z 3 30x 3 z x 4 y 5 +50x 3 y 6 +50x 2 y s 4 t 3 +62s 3 t 4 +40s 2 t x 3 +9x 2 30x 44. 6v 4 +2v 3 20v u 6 +34u 5 +30u a 4 +29a 3 +30a a 4 c 4 35a 3 c 5 +25a 2 c x 6 z 5 39x 5 z 6 +18x 4 z 7 In Exercises 49-56, factor each of the given polynomials completely y 5 +15y 4 108y 3 135y b 8 +12b 7 324b 6 432b x 6 z 5 +6x 5 z 6 144x 4 z 7 96x 3 z u 7 w 3 +9u 6 w 4 432u 5 w 5 324u 4 w z z 5 2z 4 3z x x 6 6x 5 9x a 6 c a 5 c 4 4a 4 c 5 10a 3 c a 8 c 4 +32a 7 c 5 3a 6 c 6 2a 5 c 7 In Exercises 57-60, use your calculator to help factor each of the given trinomials. Follow the procedure outline in Using the Calculator to Assist the ac-method on page x 2 +61x x 2 62x x 2 167x x 2 +x 144 Answers 1. 4y 2 z 2 (11y+6z)(11y 6z) 3. 3x 5 z 5 (x+11)(x 11) 5. 2u 5 (u+9)(u 9) 7. 3v 4 (v 2 +25)(v +5)(v 5) 9. 3x 4 (x+10)(x 10) 11. 2u 3 w 3 (25u 2 +w 2 )(5u+w)(5u w) 13. 3a 4 (5a 7) a 3 b 3 (2a+3b) b 3 (b+1) z 2 (z 1) x 2 (9x+5) b 2 c 5 (5b 8c) a 3 (a 6)(a+7)
11 466 CHAPTER 6. FACTORING 27. 3y 4 (y 8)(y 5) 29. 3z 2 (z 5)(z +9) 31. 4a 4 (a+9)(a+7) 33. 3z 2 (z +7)(z +4) 35. 5z 5 (z 6)(z 9) 37. 2b(2b 5)(b 3) 39. u 2 w 5 (u 4w)(2u+5w) 41. 2x 2 y 5 (3x+5y)(2x+5y) 43. 3x(4x 5)(x+2) 45. 2u 4 (4u+5)(u+3) 47. a 2 c 4 (4a 5c)(3a 5c) 49. 3y 2 (y +3)(y 3)(4y +5) 51. 3x 3 z 5 (x+4z)(x 4z)(3x+2z) 53. z 3 (6z +1)(6z 1)(2z +3) 55. 2a 3 c 3 (6a+c)(6a c)(2a+5c) 57. (2x+15)(3x+8) 59. (15x 8)(4x 9)
6.5 Factoring Special Forms
440 CHAPTER 6. FACTORING 6.5 Factoring Special Forms In this section we revisit two special product forms that we learned in Chapter 5, the first of which was squaring a binomial. Squaring a binomial.
FACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c
Tallahassee Community College 55 FACTORING TRINOMIALS IN THE FORM OF ax 2 + bx + c This kind of trinomial differs from the previous kind we have factored because the coefficient of x is no longer "1".
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.
Chapter R.4 Factoring Polynomials
Chapter R.4 Factoring Polynomials Introduction to Factoring To factor an expression means to write the expression as a product of two or more factors. Sample Problem: Factor each expression. a. 15 b. x
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.
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
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
Factoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1)
Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if
Factoring Guidelines. Greatest Common Factor Two Terms Three Terms Four Terms. 2008 Shirley Radai
Factoring Guidelines Greatest Common Factor Two Terms Three Terms Four Terms 008 Shirley Radai Greatest Common Factor 008 Shirley Radai Factoring by Finding the Greatest Common Factor Always check for
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
Factoring Flow Chart
Factoring Flow Chart greatest common factor? YES NO factor out GCF leaving GCF(quotient) how many terms? 4+ factor by grouping 2 3 difference of squares? perfect square trinomial? YES YES NO NO a 2 -b
Name Intro to Algebra 2. Unit 1: Polynomials and Factoring
Name Intro to Algebra 2 Unit 1: Polynomials and Factoring Date Page Topic Homework 9/3 2 Polynomial Vocabulary No Homework 9/4 x In Class assignment None 9/5 3 Adding and Subtracting Polynomials Pg. 332
Greatest Common Factor (GCF) Factoring
Section 4 4: Greatest Common Factor (GCF) Factoring The last chapter introduced the distributive process. The distributive process takes a product of a monomial and a polynomial and changes the multiplication
Factoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
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.
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
6.3 FACTORING ax 2 bx c WITH a 1
290 (6 14) Chapter 6 Factoring e) What is the approximate maximum revenue? f) Use the accompanying graph to estimate the price at which the revenue is zero. y Revenue (thousands of dollars) 300 200 100
Factoring Polynomials
Factoring Polynomials 4-1-2014 The opposite of multiplying polynomials is factoring. Why would you want to factor a polynomial? Let p(x) be a polynomial. p(c) = 0 is equivalent to x c dividing p(x). Recall
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
Tool 1. Greatest Common Factor (GCF)
Chapter 4: Factoring Review Tool 1 Greatest Common Factor (GCF) This is a very important tool. You must try to factor out the GCF first in every problem. Some problems do not have a GCF but many do. When
( ) 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
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 Algebra- Chapter 8B Assignment Sheet
Name: Factoring Algebra- Chapter 8B Assignment Sheet Date Section Learning Targets Assignment Tues 2/17 Find the prime factorization of an integer Find the greatest common factor (GCF) for a set of monomials.
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
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
The Greatest Common Factor; Factoring by Grouping
296 CHAPTER 5 Factoring and Applications 5.1 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.
5 means to write it as a product something times something instead of a sum something plus something plus something.
Intermediate algebra Class notes Factoring Introduction (section 6.1) Recall we factor 10 as 5. Factoring something means to think of it as a product! Factors versus terms: terms: things we are adding
Chapter 5. Rational Expressions
5.. Simplify Rational Expressions KYOTE Standards: CR ; CA 7 Chapter 5. Rational Expressions Definition. A rational expression is the quotient P Q of two polynomials P and Q in one or more variables, where
Factoring Polynomials
Factoring a Polynomial Expression Factoring a polynomial is expressing the polynomial as a product of two or more factors. Simply stated, it is somewhat the reverse process of multiplying. To factor polynomials,
Factoring (pp. 1 of 4)
Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common
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
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
Factoring. Factoring Monomials Monomials can often be factored in more than one way.
Factoring Factoring is the reverse of multiplying. When we multiplied monomials or polynomials together, we got a new monomial or a string of monomials that were added (or subtracted) together. For example,
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
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
CHAPTER 7: FACTORING POLYNOMIALS
CHAPTER 7: FACTORING POLYNOMIALS FACTOR (noun) An of two or more quantities which form a product when multiplied together. 1 can be rewritten as 3*, where 3 and are FACTORS of 1. FACTOR (verb) - To factor
Using the ac Method to Factor
4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trial-and-error
Factoring Trinomials of the Form x 2 bx c
4.2 Factoring Trinomials of the Form x 2 bx c 4.2 OBJECTIVES 1. Factor a trinomial of the form x 2 bx c 2. Factor a trinomial containing a common factor NOTE The process used to factor here is frequently
Factoring Quadratic Expressions
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
FACTORING ax 2 bx c WITH a 1
296 (6 20) Chapter 6 Factoring 6.4 FACTORING a 2 b c WITH a 1 In this section The ac Method Trial and Error Factoring Completely In Section 6.3 we factored trinomials with a leading coefficient of 1. In
6.4 Special Factoring Rules
6.4 Special Factoring Rules OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor a sum of cubes. By reversing the rules for multiplication
Factoring a Difference of Two Squares. Factoring a Difference of Two Squares
284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this
The majority of college students hold credit cards. According to the Nellie May
CHAPTER 6 Factoring Polynomials 6.1 The Greatest Common Factor and Factoring by Grouping 6. Factoring Trinomials of the Form b c 6.3 Factoring Trinomials of the Form a b c and Perfect Square Trinomials
PERFECT SQUARES AND FACTORING EXAMPLES
PERFECT SQUARES AND FACTORING EXAMPLES 1. Ask the students what is meant by identical. Get their responses and then explain that when we have two factors that are identical, we call them perfect squares.
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
15.1 Factoring Polynomials
LESSON 15.1 Factoring Polynomials Use the structure of an expression to identify ways to rewrite it. Also A.SSE.3? ESSENTIAL QUESTION How can you use the greatest common factor to factor polynomials? EXPLORE
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!
SPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the
FACTORING POLYNOMIALS
296 (5-40) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated
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
Polynomials and Factoring
7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of
By reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.
SECTION 5.4 Special Factoring Techniques 317 5.4 Special Factoring Techniques OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor
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
x 4-1 = (x²)² - (1)² = (x² + 1) (x² - 1) = (x² + 1) (x - 1) (x + 1)
Factoring Polynomials EXAMPLES STEP 1 : Greatest Common Factor GCF Factor out the greatest common factor. 6x³ + 12x²y = 6x² (x + 2y) 5x - 5 = 5 (x - 1) 7x² + 2y² = 1 (7x² + 2y²) 2x (x - 3) - (x - 3) =
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
7-2 Factoring by GCF. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1
7-2 Factoring by GCF Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Simplify. 1. 2(w + 1) 2. 3x(x 2 4) 2w + 2 3x 3 12x Find the GCF of each pair of monomials. 3. 4h 2 and 6h 2h 4. 13p and 26p
Factors and Products
CHAPTER 3 Factors and Products What You ll Learn use different strategies to find factors and multiples of whole numbers identify prime factors and write the prime factorization of a number find square
Operations with Algebraic Expressions: Multiplication of Polynomials
Operations with Algebraic Expressions: Multiplication of Polynomials The product of a monomial x monomial To multiply a monomial times a monomial, multiply the coefficients and add the on powers with the
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra - Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of
Introduction Assignment
PRE-CALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying
5.1 FACTORING OUT COMMON FACTORS
C H A P T E R 5 Factoring he sport of skydiving was born in the 1930s soon after the military began using parachutes as a means of deploying troops. T Today, skydiving is a popular sport around the world.
Mathematics Placement
Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.
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
7-6. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content
7-6 Choosing a Factoring Model Extension: Factoring Polynomials with More Than One Variable Essential question: How can you factor polynomials with more than one variable? What is the connection between
Polynomials. Key Terms. quadratic equation parabola conjugates trinomial. polynomial coefficient degree monomial binomial GCF
Polynomials 5 5.1 Addition and Subtraction of Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 5.3 Division of Polynomials Problem Recognition Exercises Operations on Polynomials
Academic Success Centre
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 - Grouping
6.2 Factoring - Grouping Objective: Factor polynomials with four terms using grouping. The first thing we will always do when factoring is try to factor out a GCF. This GCF is often a monomial like in
Factoring Special Polynomials
6.6 Factoring Special Polynomials 6.6 OBJECTIVES 1. Factor the difference of two squares 2. Factor the sum or difference of two cubes In this section, we will look at several special polynomials. These
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III
Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial
Algebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
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,
Factoring. Factoring Polynomial Equations. Special Factoring Patterns. Factoring. Special Factoring Patterns. Special Factoring Patterns
Factoring Factoring Polynomial Equations Ms. Laster Earlier, you learned to factor several types of quadratic expressions: General trinomial - 2x 2-5x-12 = (2x + 3)(x - 4) Perfect Square Trinomial - x
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
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
Factor Polynomials Completely
9.8 Factor Polynomials Completely Before You factored polynomials. Now You will factor polynomials completely. Why? So you can model the height of a projectile, as in Ex. 71. Key Vocabulary factor by grouping
Factoring, Solving. Equations, and Problem Solving REVISED PAGES
05-W4801-AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring
Factoring - Factoring Special Products
6.5 Factoring - Factoring Special Products Objective: Identify and factor special products including a difference of squares, perfect squares, and sum and difference of cubes. When factoring there are
In the above, the number 19 is an example of a number because its only positive factors are one and itself.
Math 100 Greatest Common Factor and Factoring by Grouping (Review) Factoring Definition: A factor is a number, variable, monomial, or polynomial which is multiplied by another number, variable, monomial,
COLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
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
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,...}
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.
When factoring, we look for greatest common factor of each term and reverse the distributive property and take out the GCF.
Factoring: reversing the distributive property. The distributive property allows us to do the following: When factoring, we look for greatest common factor of each term and reverse the distributive property
4.4 Factoring ax 2 + bx + c
4.4 Factoring ax 2 + bx + c From the last section, we now know a trinomial should factor as two binomials. With this in mind, we need to look at how to factor a trinomial when the leading coefficient is
a. You can t do the simple trick of finding two integers that multiply to give 6 and add to give 5 because the a (a = 4) is not equal to one.
FACTORING TRINOMIALS USING THE AC METHOD. Factoring trinomial epressions in one unknown is an important skill necessary to eventually solve quadratic equations. Trinomial epressions are of the form a 2
Polynomial Expression
DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL EXPRESSIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
Factoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
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
Pre-Calculus II Factoring and Operations on Polynomials
Factoring... 1 Polynomials...1 Addition of Polynomials... 1 Subtraction of Polynomials...1 Multiplication of Polynomials... Multiplying a monomial by a monomial... Multiplying a monomial by a polynomial...
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
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 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
Algebra 1 Chapter 08 review
Name: Class: Date: ID: A Algebra 1 Chapter 08 review Multiple Choice Identify the choice that best completes the statement or answers the question. Simplify the difference. 1. (4w 2 4w 8) (2w 2 + 3w 6)
SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen
SOLVING QUADRATIC EQUATIONS - COMPARE THE FACTORING ac METHOD AND THE NEW DIAGONAL SUM METHOD By Nghi H. Nguyen A. GENERALITIES. When a given quadratic equation can be factored, there are 2 best methods
Factoring Trinomials of the Form
Section 4 6B: Factoring Trinomials of the Form A x 2 + Bx + C where A > 1 by The AC and Factor By Grouping Method Easy Trinomials: 1 x 2 + Bx + C The last section covered the topic of factoring second
FACTOR POLYNOMIALS by SPLITTING
FACTOR POLYNOMIALS by SPLITTING THE IDEA FACTOR POLYNOMIALS by SPLITTING The idea is to split the middle term into two pieces. Say the polynomial looks like c + bx + ax 2. Further more suppose the DL METHOD
2x 2x 2 8x. Now, let s work backwards to FACTOR. We begin by placing the terms of the polynomial inside the cells of the box. 2x 2
Activity 23 Math 40 Factoring using the BOX Team Name (optional): Your Name: Partner(s): 1. (2.) Task 1: Factoring out the greatest common factor Mini Lecture: Factoring polynomials is our focus now. Factoring
GCF/ Factor by Grouping (Student notes)
GCF/ Factor by Grouping (Student notes) Factoring is to write an expression as a product of factors. For example, we can write 10 as (5)(2), where 5 and 2 are called factors of 10. We can also do this