By reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.


 Janis Marshall
 3 years ago
 Views:
Transcription
1 SECTION 5.4 Special Factoring Techniques Special Factoring Techniques 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 of binomials from Section 4.6, we get rules for factoring polynomials in certain forms. OBJECTIVE 1 Factor a difference of squares. The formula for the product of the sum and difference of the same two terms is Factoring a Difference of Squares 1x + y21x  y2 = x 2  y 2. Reversing this rule leads to the following special factoring rule. x 2 y 2 1x y21x y2 For example, The following conditions must be true for a binomial to be a difference of squares. 1. Both terms of the binomial must be squares, such as x 2, 9y 2 =13y2 2, m 216 = m =1m + 421m = 5 2, 1 = 1 2, m 4 =1m The terms of the binomial must have different signs (one positive and one negative). EXERCISE 1 Factor each binomial if possible. x x EXAMPLE 1 Factoring Differences of Squares Factor each binomial if possible. x 2  y 2 y21x a 249 = a =1a + 721a  72 = 1x + x 28 Because 8 is not the square of an integer, this binomial does not satisfy the conditions above. It is a prime polynomial. (d) p Since p is a sum of squares, it is not equal to 1 p p Also, we use FOIL and try the following. 1 p p p p Thus, p is a prime polynomial. y2 = p 28p + 16, not p = p 2 + 8p + 16, not p y 2  m 2 =1 y + m21 y  m2 ANSWERS 1. 1x x prime CAUTION As Example 1(d) suggests, after any common factor is removed, a sum of squares cannot be factored.
2 318 CHAPTER 5 Factoring and Applications EXERCISE 2 Factor each difference of squares. 9t a 249b 2 EXAMPLE 2 Factoring Differences of Squares Factor each difference of squares. 25m 216 =15m =15m m z 264t 2 x 2 =17z2 218t2 2  =17z + 8t217z  8t2 y 2 = 1x + y21x y2 Write each term as a square. Factor the difference of squares.  NOTE Always check a factored form by multiplying. EXERCISE 3 Factor completely. 16k 264 m v EXAMPLE 3 Factor completely. 81y 236 = 919y 242 = 9313y Factoring More Complex Differences of Squares = 913y y  22 Neither binomial can be factored further. Don t stop here. m 416 p 436 =1m =1 p =1 p p 262 =1m m 242 =1m m + 221m  22 Factor out the GCF, 9. Write each term as a square. Factor the difference of squares. Write each term as a square. Factor the difference of squares. Factor the difference of squares. Factor the difference of squares again. CAUTION Factor again when any of the factors is a difference of squares, as in Example 3. Check by multiplying. ANSWERS 2. 13t t a + 7b216a  7b k + 221k m m v v + 521v  52 OBJECTIVE 2 Factor a perfect square trinomial. The expressions 144, 4x 2, and 81m 6 are called perfect squares because 144 = 12 2, 4x 2 =12x2 2, and 81m 6 =19m A perfect square trinomial is a trinomial that is the square of a binomial. For example, x 2 + 8x + 16 is a perfect square trinomial because it is the square of the binomial x + 4. x 2 + 8x + 16 =1x + 421x + 42 =1x
3 SECTION 5.4 Special Factoring Techniques 319 On the one hand, a necessary condition for a trinomial to be a perfect square is that two of its terms be perfect squares. For this reason, 16x 2 + 4x + 15 is not a perfect square trinomial, because only the term 16x 2 is a perfect square. On the other hand, even if two of the terms are perfect squares, the trinomial may not be a perfect square trinomial. For example, x 2 + 6x + 36 has two perfect square terms, x 2 and 36, but it is not a perfect square trinomial. Factoring Perfect Square Trinomials x 2 2xy y 2 1x y2 2 x 2 2xy y 2 1x y2 2 The middle term of a perfect square trinomial is always twice the product of the two terms in the squared binomial (as shown in Section 4.6). Use this rule to check any attempt to factor a trinomial that appears to be a perfect square. EXERCISE 4 Factor y y EXAMPLE 4 Factoring a Perfect Square Trinomial Factor x x The x 2 term is a perfect square, and so is 25. Try to factor x x + 25 as 1x To check, take twice the product of the two terms in the squared binomial. 2 # x # 5 = 10x Middle term of x x + 25 Twice First term Last term of binomial of binomial Since 10x is the middle term of the trinomial, the trinomial is a perfect square. x x + 25 factors as 1x EXAMPLE 5 Factoring Perfect Square Trinomials Factor each trinomial. x 222x The first and last terms are perfect squares 1121 = 11 2 or Check to see whether the middle term of x 222x is twice the product of the first and last terms of the binomial x # x #1112 = 22x Middle term of x 222x Twice First Last term term Thus, x 222x is a perfect square trinomial. x 222x factors as 1x Same sign ANSWER 4. 1 y Notice that the sign of the second term in the squared binomial is the same as the sign of the middle term in the trinomial.
4 320 CHAPTER 5 Factoring and Applications EXERCISE 5 Factor each trinomial. t 218t p 228p x 2 + 6x + 4 (d) 80x x x 9m 224m + 16 =13m m =13m Twice First Last term term 25y y + 16 The first and last terms are perfect squares. and Twice the product of the first and last terms of the binomial 5y + 4 is which is not the middle term of This trinomial is not a perfect square. In fact, the trinomial cannot be factored even with the methods of the previous sections. It is a prime polynomial. (d) 12z z z = 3z14z z y 2 =15y2 2 2 # 5y # 4 = 40y, 25y y = 4 2 Factor out the common factor, 3z. = 3z312z z z2 + 20z + 25 is a perfect square trinomial. = 3z12z Factor. NOTE 1. The sign of the second term in the squared binomial is always the same as the sign of the middle term in the trinomial. 2. The first and last terms of a perfect square trinomial must be positive, because they are squares. For example, the polynomial x 22x  1 cannot be a perfect square, because the last term is negative. 3. Perfect square trinomials can also be factored by using grouping or the FOIL method, although using the method of this section is often easier. OBJECTIVE 3 Factor a difference of cubes. We can factor a difference of cubes by using the following pattern. Factoring a Difference of Cubes x 3 y 3 1x y21x 2 xy y 2 2 ANSWERS 5. 1t p prime (d) 5x14x This pattern for factoring a difference of cubes should be memorized. To see that the pattern is correct, multiply 1x  y21x 2 + xy + y 2 2. x 2 + xy + y 2 x  y Multiply vertically. (Section 4.5) y1x 2 + xy + y x 2 y  xy 2  y 3 x 3 + x 2 y + xy 2 x1x2 + xy + y 2 2 x 3  y 3 Add.
5 SECTION 5.4 Special Factoring Techniques 321 Notice the pattern of the terms in the factored form of x 3  y 3. x 3  y 3 = (a binomial factor)(a trinomial factor) The binomial factor has the difference of the cube roots of the given terms. The terms in the trinomial factor are all positive. The terms in the binomial factor help to determine the trinomial factor. x 3  y 3 =1x  y21 positive First term product of second term squared + the terms + squared x 2 + xy + y 2 2 CAUTION The polynomial x 3  y 3 is not equivalent to 1x  y2 3. x 3  y 3 1x  y2 3 =1x  y21x 2 + xy + y 2 2 =1x  y21x  y21x  y2 =1x  y21x 22xy + y 2 2 EXERCISE 6 Factor each polynomial. a t k (d) 125x 3343y 6 ANSWERS 6. 1a  321a 2 + 3a t t t k  421k 2 + 4k (d) 15x  7y 2 2 # 125x xy y 4 2 EXAMPLE 6 Factor each polynomial. (d) Factoring Differences of Cubes m Let x = m and y = 5 in the pattern for the difference of cubes. m = m =1m  521m 2 + 5m p 327 =12p x 3 y 3 =1m  521m 2 + 5m =12p p p =12p p 2 + 6p m 332 = 41m 382 = 41m t 3216s 6 =15t2 316s = 1x  y21x p2 2 = 2 2 p 2 = 4p 2, NOT 2p 2. = 41m  221m 2 + 2m + 42 =15t  6s t t16s s =15t  6s t ts s 4 2 xy + y 2 2 8p 3 =12p2 3 and 27 = 3 3. Let x = 2p, y = 3. Let x = m, y = = 25 Apply the exponents. Multiply. Factor out the common factor, 4. 8 = 2 3 Factor the difference of cubes. Write each term as a cube. Factor the difference of cubes. Apply the exponents. Multiply.
6 322 CHAPTER 5 Factoring and Applications CAUTION A common error in factoring a difference of cubes, such as x 3  y 3 =1x  y21x 2 + xy + y 2 2, is to try to factor x 2 + xy + y 2. This is usually not possible. OBJECTIVE 4 Factor a sum of cubes. A sum of squares, such as cannot be factored by using real numbers, but a sum of cubes can. m , Factoring a Sum of Cubes x 3 y 3 1x y21x 2 xy y 2 2 Compare the pattern for the sum of cubes with that for the difference of cubes. Positive x 3  y 3 =1x  y21x 2 + xy + y 2 2 Difference of cubes Same sign Opposite sign Positive The only difference between the patterns is the positive and negative signs. x 3 + y 3 =1x + y21x 2  xy + y 2 2 Sum of cubes Same sign Opposite sign EXERCISE 7 Factor each polynomial. x a 3 + 8b 3 ANSWERS 7. 1x + 521x 25x a + 2b219a 26ab + 4b 2 2 EXAMPLE 7 Factoring Sums of Cubes Factor each polynomial. k = k = 3 3 =1k + 321k 23k Factor the sum of cubes. =1k + 321k 23k + 92 Apply the exponent. 8m n 3 =12m n2 3 8m 3 =12m2 3 and 125n 3 =15n2 3. =12m + 5n2312m2 22m15n2 +15n2 2 4 Factor the sum of cubes. =12m + 5n214m 210mn + 25n a b 3 =110a b2 3 Be careful: 12m2 2 = 2 2 m 2 and 15n2 2 = 5 2 n 2. =110a 2 + 3b23110a a 2 213b2 +13b2 2 4 Factor the sum of cubes. =110a 2 + 3b21100a 430a 2 b + 9b a = a = 100a 4
7 SECTION 5.4 Special Factoring Techniques 323 The methods of factoring discussed in this section are summarized here. Special Factorizations Difference of squares x 2 y 2 1x y21x y2 Perfect square trinomials x 2 2xy y 2 1x y2 2 x 2 2xy y 2 1x y2 2 Difference of cubes x 3 y 3 1x y21x 2 xy y 2 2 Sum of cubes x 3 y 3 1x y21x 2 xy y 2 2 The sum of squares can be factored only if the terms have a common factor. 5.4 EXERCISES Complete solution available on the Video Resources on DVD 1. Concept Check To help you factor the difference of squares, complete the following list of squares. 1 2 = 6 2 = 11 2 = 16 2 = 17 2 = 18 2 = 19 2 = 20 2 = 2. Concept Check The following powers of x are all perfect squares: x 2, x 4, x 6, x 8, x 10. On the basis of this observation, we may make a conjecture (an educated guess) that if the power of a variable is divisible by (with 0 remainder), then we have a perfect square. 3. Concept Check To help you factor the sum or difference of cubes, complete the following list of cubes. 1 3 = 2 2 = 7 2 = 12 2 = 2 3 = 3 2 = 8 2 = 13 2 = 3 3 = 4 2 = 9 2 = 14 2 = 4 3 = 5 2 = 10 2 = 15 2 = 5 3 = 6 3 = 7 3 = 8 3 = 9 3 = 10 3 = 4. Concept Check The following powers of x are all perfect cubes: x 3, x 6, x 9, x 12, x 15. On the basis of this observation, we may make a conjecture that if the power of a variable is divisible by (with 0 remainder), then we have a perfect cube. 5. Concept Check Identify each monomial as a perfect square, a perfect cube, both of these, or neither of these. 64x 6 y t 6 49x 12 (d) 81r Concept Check What must be true for x n to be both a perfect square and a perfect cube? Factor each binomial completely. If the binomial is prime, say so. Use your answers from Exercises 1 and 2 as necessary. See Examples y t x x m k m x r x x a 28
8 324 CHAPTER 5 Factoring and Applications p q r 225a m 2100p x w p r x y 410, p k Concept Check When a student was directed to factor k 481 from Exercise 30 completely, his teacher did not give him full credit for the answer 1k k The student argued that since his answer does indeed give k 481 when multiplied out, he should be given full credit. WHAT WENT WRONG? Give the correct factored form. 32. Concept Check The binomial 4x is a sum of squares that can be factored. How is this binomial factored? When can the sum of squares be factored? Concept Check Find the value of the indicated variable. 33. Find b so that x 2 + bx + 25 factors as 1x Find c so that 4m 212m + c factors as 12m Find a so that ay 212y + 4 factors as 13y Find b so that 100a 2 + ba + 9 factors as 110a Factor each trinomial completely. See Examples 4 and w 2 + 2w p 2 + 4p x 28x x 210x x x y y x 240x y 260y x 228xy + 4y z 212zw + 9w x xy + 9y t tr + 16r h 240hy + 8y x 248xy + 32y k 34k 2 + 9k 52. 9r 36r r z 4 + 5z 3 + z x 4 + 2x 3 + x 2 Factor each binomial completely. Use your answers from Exercises 3 and 4 as necessary. See Examples 6 and a m m b k p x y p y w 3216z x x y 38x x 316y w 3216z p q x y a b m 3 + 8p t 3 + 8s r s x 3125y t 364s m 6 + 8n r s x 9 + y x 9  y 9
9 Summary Exercises on Factoring 325 Although we usually factor polynomials using integers, we can apply the same concepts to factoring using fractions and decimals. z = z 2  a 3 4 b = A3 4B 2 = az baz b Factor the difference of squares. Apply the special factoring rules of this section to factor each binomial or trinomial. 83. p 84. q b x y t 90. m x 21.0x m t y 93. x y Brain Busters Factor each polynomial completely. x m + n2 21m  n a  b2 31a + b m 2  p 2 + 2m + 2p 98. 3r  3k + 3r 23k 2 36m PREVIEW EXERCISES Solve each equation. See Sections 2.1 and m  4 = t + 2 = t + 10 = x = 0 SUMMARY EXERCISES on Factoring As you factor a polynomial, ask yourself these questions to decide on a suitable factoring technique. Factoring a Polynomial 1. Is there a common factor? If so, factor it out. 2. How many terms are in the polynomial? Two terms: Check to see whether it is a difference of squares or a sum or difference of cubes. If so, factor as in Section 5.4. Three terms: Is it a perfect square trinomial? If the trinomial is not a perfect square, check to see whether the coefficient of the seconddegree term is 1. If so, use the method of Section 5.2. If the coefficient of the seconddegree term of the trinomial is not 1, use the general factoring methods of Section 5.3. Four terms: Try to factor the polynomial by grouping, as in Section Can any factors be factored further? If so, factor them. (continued)
10 326 CHAPTER 5 Factoring and Applications Match each polynomial in Column I with the best choice for factoring it in Column II. The choices in Column II may be used once, more than once, or not at all x x x 217x + 72 I 3. 16m 2 n + 24mn  40mn a 2121b p 260pq + 25q 2 6. z 24z r x 6 + 4x 43x w z 224z Factor each polynomial completely. II A. Factor out the GCF. No further factoring is possible. B. Factor a difference of squares. C. Factor a difference of cubes. D. Factor a sum of cubes. E. Factor a perfect square trinomial. F. Factor by grouping. G. Factor out the GCF. Then factor a trinomial by grouping or trial and error. H. Factor into two binomials by finding two integers whose product is the constant in the trinomial and whose sum is the coefficient of the middle term. I. The polynomial is prime. 11. a 24a a a y 26y y y 5168y a + 12b + 18c 16. m 23mn  4n p 217p z 26z + 7z z 27z m 210m x 3 y xy y a 58a 448a k 210k z 23za  10a z x 24x  5x n 2 r nr 350n 2 r 29. 6n 219n y y x m 2 + 2m y 25y m z z x x k 212k p p m 224z m 22m k 2 + 4k a 3 b 560a 4 b a 6 b k 3 + 7k 270k r  5s  rs 45. y y 530y m  16m k z y 2  y k p 1045p 9252p m m m m r rm + 9m z 212z h hg  14g z 345z z 59. k 211k p 2100m 2
11 Summary Exercises on Factoring k 312k 215k 62. y 24yk  12k p r m + 2p + mp 66. 2m 2 + 7mn  15n z 28z m 4400m 3 n + 195m 2 n m 236m a 281y x 2  xy + y y z z 216z m m m + 12n + 3mn q  6p + 3pq 77. 6a a y 642y 5120y a 3  b 3 + 2a  2b k 248k m 280mn + 25n y 3 z y 224y 4 z k 22kh  3h a 27a x a y 27yz  6z m 24m a ab  3b a RELATING CONCEPTS EXERCISES FOR INDIVIDUAL OR GROUP WORK A binomial may be both a difference of squares and a difference of cubes. One example of such a binomial is x 61. With the techniques of Section 5.4, one factoring method will give the completely factored form, while the other will not. Work Exercises in order to determine the method to use if you have to make such a decision. 91. Factor x 61 as the difference of squares. 92. The factored form obtained in Exercise 91 consists of a difference of cubes multiplied by a sum of cubes. Factor each binomial further. 93. Now start over and factor x 61 as the difference of cubes. 94. The factored form obtained in Exercise 93 consists of a binomial that is a difference of squares and a trinomial. Factor the binomial further. 95. Compare your results in Exercises 92 and 94. Which one of these is factored completely? 96. Verify that the trinomial in the factored form in Exercise 94 is the product of the two trinomials in the factored form in Exercise Use the results of Exercises to complete the following statement: In general, if I must choose between factoring first with the method for the difference of squares or the method for the difference of cubes, I should choose the method to eventually obtain the completely factored form. 98. Find the completely factored form of x by using the knowledge you gained in Exercises
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
More information1.3 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.
More informationFactoring 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
More informationNSM100 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
More informationThe 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.
More informationPolynomials 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
More informationFACTORING POLYNOMIALS
296 (540) 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
More informationFactoring 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
More informationIn algebra, factor by rewriting a polynomial as a product of lowerdegree 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
More informationGreatest 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
More informationFACTORING ax 2 bx c. Factoring Trinomials with Leading Coefficient 1
5.7 Factoring ax 2 bx c (549) 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.
More informationUsing 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 trialanderror
More information6.1 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.
More informationexpression 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
More information( ) 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
More informationFactoring 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
More informationPolynomials. 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
More informationFactoring 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,
More informationFactoring 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
More informationTool 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
More informationOperations 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
More informationFactoring 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
More informationFactoring 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.
More informationFactoring (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
More informationFactoring 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
More informationFACTORING 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
More informationFactoring 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
More informationDefinitions 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
More informationChapter 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
More informationFACTORING 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".
More informationMATH 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
More informationAlum 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
More informationFactoring. 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,
More informationSection 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
More informationPOLYNOMIALS 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
More informationThis 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 byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationAIP 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
More informationA. 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!
More informationFactoring 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
More informationFactoring 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
More information76. Choosing a Factoring Model. Extension: Factoring Polynomials with More Than One Variable IN T RO DUC E T EACH. Standards for Mathematical Content
76 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
More informationName 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
More information1.3 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,
More informationCopy 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,...}
More informationA 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
More informationFactoring 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
More information6.1 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
More information6.3 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
More informationSimplification of Radical Expressions
8. Simplification of Radical Expressions 8. OBJECTIVES 1. Simplify a radical expression by using the product property. Simplify a radical expression by using the quotient property NOTE A precise set of
More informationAlgebra 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
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More information5.1 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.
More informationFactoring  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
More informationWhen 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
More informationThe 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
More information72 Factoring by GCF. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1
72 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
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationAcademic Success Centre
250) 9606367 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
More informationRadicals  Multiply and Divide Radicals
8. Radicals  Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals
More informationMATH 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.
More informationFactoring. 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 25x12 = (2x + 3)(x  4) Perfect Square Trinomial  x
More informationSimplifying Algebraic Fractions
5. Simplifying Algebraic Fractions 5. OBJECTIVES. Find the GCF for two monomials and simplify a fraction 2. Find the GCF for two polynomials and simplify a fraction Much of our work with algebraic fractions
More information15.1 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
More informationAlgebra 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)
More informationMATH 102 College Algebra
FACTORING Factoring polnomials ls is simpl the reverse process of the special product formulas. Thus, the reverse process of special product formulas will be used to factor polnomials. To factor polnomials
More informationFactoring 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
More informationDeterminants 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
More informationEAP/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
More informationFactoring, Solving. Equations, and Problem Solving REVISED PAGES
05W4801AM1.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
More informationVeterans 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
More informationWelcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move
More informationFactor 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
More informationx 41 = (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) =
More informationSECTION 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
More informationMath 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
More informationPERFECT 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.
More informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
More informationFactors 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
More informationAlgebra Cheat Sheets
Sheets Algebra Cheat Sheets provide you with a tool for teaching your students notetaking, problemsolving, and organizational skills in the context of algebra lessons. These sheets teach the concepts
More informationLesson 9: Radicals and Conjugates
Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.
More informationSect 6.1  Greatest Common Factor and Factoring by Grouping
Sect 6.1  Greatest Common Factor and Factoring by Grouping Our goal in this chapter is to solve nonlinear equations by breaking them down into a series of linear equations that we can solve. To do this,
More informationMath 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
More informationFACTORING 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
More informationCHAPTER 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
More informationTopic: Special Products and Factors Subtopic: Rules on finding factors of polynomials
Quarter I: Special Products and Factors and Quadratic Equations Topic: Special Products and Factors Subtopic: Rules on finding factors of polynomials Time Frame: 20 days Time Frame: 3 days Content Standard:
More informationFactoring  Greatest Common Factor
6.1 Factoring  Greatest Common Factor Objective: Find the greatest common factor of a polynomial and factor it out of the expression. The opposite of multiplying polynomials together is factoring polynomials.
More informationPreCalculus 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...
More informationMathematics Curriculum
Common Core Mathematics Curriculum Table of Contents 1 Polynomial and Quadratic Expressions, Equations, and Functions MODULE 4 Module Overview... 3 Topic A: Quadratic Expressions, Equations, Functions,
More informationFactoring  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
More informationAlgebra 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
More informationHIBBING 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,
More informationFactoring Polynomials
Factoring Polynomials 412014 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
More informationFactoring 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
More information#6 Opener Solutions. Move one more spot to your right. Introduce yourself if needed.
1. Sit anywhere in the concentric circles. Do not move the desks. 2. Take out chapter 6, HW/notes #1#5, a pencil, a red pen, and your calculator. 3. Work on opener #6 with the person sitting across from
More informationSPECIAL PRODUCTS AND FACTORS
SPECIAL PRODUCTS AND FACTORS I. INTRODUCTION AND FOCUS QUESTIONS http://dmciresidences.com/home/20/0/ cedarcrestcondominiums/ http://frontiernerds.com/metalbox http://mazharalticonstruction.blogspot.
More informationLagrange 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
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More information