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1、? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 418 Chapter 8 Factoring Factoring 8 Real-World Link Dolphins Factoring is used to solve problems involving vertical motion. For example, factoring can be used to determine how long a dolphin that jumps out of the water is in the air. 1Fold in thirds and then in half alo
2、ng the width. 3Open. Cut short side along folds to make tabs. 2Open. Fold lengthwise, leaving a 1 _ 2 ” tab on the right. 4Label each tab as shown. Factoring Make this Foldable to help you organize your notes on factoring. Begin with a sheet of plain 8 1 _ 2 ” by 11” paper. Standard 11.0 Students ap
3、ply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. Standard 14.0 Students solve a quadratic equation by
4、factoring or completing the square. (Key) Key Vocabulary factored form (p. 421) perfect square trinomials (p. 454) prime polynomial (p. 443) 0418-0419 COCH08-877852.indd 4180418-0419 COCH08-877852.indd 4189/21/06 5:58:40 PM9/21/06 5:58:40 PM Chapter 8 Get Ready For Chapter 8 419 Rewrite each express
5、ion using the Distributive Property. Then simplify. (Lesson 1-5) 1. 3(4 - x) 2. a(a + 5) 3. -7( n 2 - 3n + 1) 4. 6y(-3y - 5y - 5y 2 + y 3 ) 5. JOBS In a typical week, Mr. Jackson averages 4 hours using e-mail, 10 hours of meeting in person, and 20 hours on the telephone. Write an expression that cou
6、ld be used to determine how many hours he will spend on these activities over the next month. Find each product. (Lesson 7-6) 6. (x + 4)(x + 7) 7. (3n - 4)(n + 5) 8. (6a - 2b)(9a + b) 9. (-x - 8y)(2x - 12y) 10. TABLE TENNIS The dimensions of a homemade table tennis table are represented by a width o
7、f 2x + 3 and a length of x + 1. Find an expression for the area of the table tennis table. Find each product. (Lesson 7-7) 11. (y + 9) 2 12. (3a - 2) 2 13. (3m + 5n ) 2 14. (6r - 7s ) 2 EXAMPLE 1 Rewrite n (n - 3n 2 + 2 + 4 _ n ) using the Distributive Property. Then simplify. n (n - 3n 2 + 2 + 4 _
8、n ) Original expression = (n)(n) + (n) (-3 n 2 ) + (n)(2) + (n) ( 4 _ n ) Distribute n to each term inside the parentheses. = n 2 - 3 n 3 + 2n + 4 Multiply. = -3n 3 + n 2 + 2n + 4 Rewrite in descending order with respect to the exponents. EXAMPLE 2 Find (x + 2)(3x - 1). (x + 2)(3x - 1) Original expr
9、ession = (x)(3x) + (x)(-1) + (2)(3x) + (2)(-1) FOIL Method = 3x 2 - x + 6x - 2 Multiply. = 3x 2 + 5x - 2 Combine like terms. EXAMPLE 3 Find (3 - g ) 2 . (3 - g ) 2 = (3 - g)(3 - g) Laws of Exponents = 3 2 - 3g - 3g + g 2 Multiply. = 3 2 - 6g + g 2 Combine like terms. = 9 - 6g + g 2 Simplify. GET REA
10、DY for Chapter 8 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Option 2 Take the Online Readiness Quiz at . 0418-0419 COCH08-877852.indd 4190418-0419 COCH08-877852.indd 41910/10/06 3:53:43 PM10/10/06
11、 3:53:43 PM 1 ? 18 2 ? 9 3 ? 6 420 Chapter 8 Factoring 8-1 Monomials and Factoring In the search for extraterrestrial life, scientists listen to radio signals coming from faraway galaxies. How can they be sure that a particular radio signal was deliberately sent by intelligent beings instead of comi
12、ng from some natural phenomenon? What if that signal began with a series of beeps in a pattern composed of the first 30 prime numbers (“beep-beep,” “beep-beep- beep,” and so on)? Prime Factorization Numbers that are multiplied are factors of the resulting product. Numbers that have whole number fact
13、ors can be represented geometrically. Consider all of the possible rectangles with whole number dimensions that have areas of 18 square units. The number 18 has six factors: 1, 2, 3, 6, 9, and 18. Prime Numbers Before deciding that a number is prime, try dividing it by all of the prime numbers that
14、are less than the square root of that number. A whole number expressed as the product of prime factors is called the prime factorization of the number. Two methods of factoring 90 are shown. Method 1 Find the least prime factors. 90 = 2 45 The least prime factor of 90 is 2. = 2 3 15 The least prime
15、factor of 45 is 3. = 2 3 3 5 The least prime factor of 15 is 3. Prime and Composite Numbers Words A whole number, greater than 1, for which the only factors are 1 and itself, is called a prime number. A whole number, greater than 1, that has more than two factors is called a composite number. Exampl
16、es 2, 3, 5, 7, 11, 13, 17, 19 4, 6, 8, 9, 10, 12, 14, 15 0 and 1 are neither prime nor composite. Main Ideas Find prime factorizations of monomials. Find the greatest common factors of monomials. Preparation for Standard 11.0 Students apply basic factoring techniques to second- and simple third- degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two
