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Warm Up Tell whether the second number is a factor of the first number 1. 50, 6 no 2. 105, 7 3. List the factors of 28. ±14, ±28 yes ±1, ±2, ±4, ±7, Tell whether each number is prime or composite. If the number is composite, write it as the product of two numbers. 4. 11 prime 5. 98 composite; 49 • 2 Objectives Write the prime factorization of numbers. Find the GCF of monomials. Vocabulary prime factorization greatest common factor The whole numbers that are multiplied to find a product are called factors of that product. A number is divisible by its factors. You can use the factors of a number to write the number as a product. The number 12 can be factored several ways. Factorizations of 12 • • • • • • • The order of factors does not change the product, but there is only one example below that cannot be factored further. The circled factorization is the prime factorization because all the factors are prime numbers. The prime factors can be written in any order, and except for changes in the order, there is only one way to write the prime factorization of a number. Factorizations of 12 • • • • • • • Remember! A prime number has exactly two factors, itself and 1. The number 1 is not prime because it only has one factor. Example 1: Writing Prime Factorizations Write the prime factorization of 98. Method 1 Factor tree Method 2 Ladder diagram Choose any two factors Choose a prime factor of 98 of 98 to begin. Keep finding to begin. Keep dividing by factors until each branch prime factors until the ends in a prime factor. quotient is 1. 98 2 98 7 49 2 49 7 7 7 7 1 98 = 2 7 7 98 = 2 7 7 The prime factorization of 98 is 2 7 7 or 2 72. Example 2 Write the prime factorization of each number. a. 40 40 2 20 2 10 2 5 40 = 23 5 The prime factorization of 40 is 2 2 2 5 or 23 5. b. 33 11 33 3 33 = 3 11 The prime factorization of 33 is 3 11. Example 3 Write the prime factorization of each number. c. 49 d. 19 49 7 7 49 = 7 7 The prime factorization of 49 is 7 7 or 72. 1 19 19 19 = 1 19 The prime factorization of 19 is 1 19. Factors that are shared by two or more whole numbers are called common factors. The greatest of these common factors is called the greatest common factor, or GCF. Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 32: 1, 2, 4, 8, 16, 32 Common factors: 1, 2, 4 The greatest of the common factors is 4. Example 4: Finding the GCF of Numbers Find the GCF of each pair of numbers. 100 and 60 Method 1 List the factors. factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100 List all the factors. factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 Circle the GCF. The GCF of 100 and 60 is 20. Example 5: Finding the GCF of Numbers Find the GCF of each pair of numbers. 26 and 52 Method 2 Prime factorization. 26 = 2 13 52 = 2 2 13 2 13 = 26 Write the prime factorization of each number. Align the common factors. The GCF of 26 and 52 is 26. Example 6 Find the GCF of each pair of numbers. 12 and 16 Method 1 List the factors. factors of 12: 1, 2, 3, 4, 6, 12 List all the factors. factors of 16: 1, 2, 4, 8, 16 Circle the GCF. The GCF of 12 and 16 is 4. Example 7 Find the GCF of each pair of numbers. 15 and 25 Method 2 Prime factorization. 15 = 1 3 5 25 = 1 5 5 1 5=5 Write the prime factorization of each number. Align the common factors. You can also find the GCF of monomials that include variables. To find the GCF of monomials, write the prime factorization of each coefficient and write all powers of variables as products. Then find the product of the common factors. Example 3A: Finding the GCF of Monomials Find the GCF of each pair of monomials. 15x3 and 9x2 15x3 = 3 5 x x x 9x2 = 3 3 x x 3 Write the prime factorization of each coefficient and write powers as products. Align the common factors. x x = 3x2 Find the product of the common factors. The GCF of 3x3 and 6x2 is 3x2. Example 3B: Finding the GCF of Monomials Find the GCF of each pair of monomials. 8x2 and 7y3 Write the prime factorization of each 8x2 = 2 2 2 xx coefficient and write 7y3 = 7 y y y powers as products. Align the common factors. The GCF 8x2 and 7y is 1. There are no common factors other than 1. Helpful Hint If two terms contain the same variable raised to different powers, the GCF will contain that variable raised to the lower power. Example 8 Find the GCF of each pair of monomials. 18g2 and 27g3 18g2 = 2 3 3 27g3 = gg Write the prime factorization of each coefficient and write powers as products. 3 3 3 g g g Align the common factors. 33 gg Find the product of the common factors. The GCF of 18g2 and 27g3 is 9g2. Example 9 Find the GCF of each pair of monomials. 16a6 and 9b 16a6 = 2 2 2 2 a a a a a a 9b = The GCF of 16a6 and 7b is 1. Write the prime factorization of each coefficient and write powers as products. 33b Align the common factors. There are no common factors other than 1. Example 10 Find the GCF of each pair of monomials. 8x and 7v2 8x = 2 2 2 x 7v2 = 7vv The GCF of 8x and 7v2 is 1. Write the prime factorization of each coefficient and write powers as products. Align the common factors. There are no common factors other than 1. Example 11 A cafeteria has 18 chocolate-milk cartons and 24 regular-milk cartons. The cook wants to arrange the cartons with the same number of cartons in each row. Chocolate and regular milk will not be in the same row. How many rows will there be if the cook puts the greatest possible number of cartons in each row? The 18 chocolate and 24 regular milk cartons must be divided into groups of equal size. The number of cartons in each row must be a common factor of 18 and 24. Example 11 Continued Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Find the common factors of 18 and 24. The GCF of 18 and 24 is 6. The greatest possible number of milk cartons in each row is 6. Find the number of rows of each type of milk when the cook puts the greatest number of cartons in each row. Example 11 Continued 18 chocolate milk cartons = 3 rows 6 containers per row 24 regular milk cartons 6 containers per row = 4 rows When the greatest possible number of types of milk is in each row, there are 7 rows in total. Example 12 Adrianne is shopping for a CD storage unit. She has 36 CDs by pop music artists and 48 CDs by country music artists. She wants to put the same number of CDs on each shelf without putting pop music and country music CDs on the same shelf. If Adrianne puts the greatest possible number of CDs on each shelf, how many shelves does her storage unit need? The 36 pop and 48 country CDs must be divided into groups of equal size. The number of CDs in each row must be a common factor of 36 and 48. Example 12 Continued Find the common Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 factors of 36 and 48. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 The GCF of 36 and 48 is 12. The greatest possible number of CDs on each shelf is 12. Find the number of shelves of each type of CDs when Adrianne puts the greatest number of CDs on each shelf. 36 pop CDs 12 CDs per shelf = 3 shelves 48 country CDs 12 CDs per shelf = 4 shelves When the greatest possible number of CD types are on each shelf, there are 7 shelves in total. Recall that the Distributive Property states that ab + ac =a(b + c). The Distributive Property allows you to “factor” out the GCF of the terms in a polynomial to write a factored form of the polynomial. A polynomial is in its factored form when it is written as a product of monomials and polynomials that cannot be factored further. The polynomial 2(3x – 4x) is not fully factored because the terms in the parentheses have a common factor of x. Example 13 : Factoring by Using the GCF Factor the polynomial. Check your answer. 2x2 – 4 2x2 = 2 xx 4=22 2 2x2 – (2 2) 2(x2 – 2) Check 2(x2 – 2) 2x2 – 4 Find the GCF. The GCF of 2x2 and 4 is 2. Write terms as products using the GCF as a factor. Use the Distributive Property to factor out the GCF. Multiply to check your answer. The product is the original polynomial. Example 14 : Factoring by Using the GCF Factor the polynomial. Check your answer. 8x3 – 4x2 – 16x 8x3 = 2 2 2 x x x Find the GCF. 4x2 = 2 2 xx 16x = 2 2 2 2 x The GCF of 8x3, 4x2, and 16x is 4x. 22 x = 4x Write terms as products using the GCF as a factor. 2x2(4x) – x(4x) – 4(4x) Use the Distributive Property to 4x(2x2 – x – 4) factor out the GCF. Check 4x(2x2 – x – 4) Multiply to check your answer. The product is the original 8x3 – 4x2 – 16x polynomials. Example 15 : Factoring by Using the GCF Factor the polynomial. Check your answer. –14x – 12x2 – 1(14x + 12x2) 14x = 2 7x 12x2 = 2 2 3 xx 2 –1[7(2x) + 6x(2x)] –1[2x(7 + 6x)] –2x(7 + 6x) Both coefficients are negative. Factor out –1. Find the GCF. 2 The GCF of 14x and 12x x = 2x is 2x. Write each term as a product using the GCF. Use the Distributive Property to factor out the GCF. Example 16: Factoring by Using the GCF Factor the polynomial. Check your answer. –14x – 12x2 Check –2x(7 + 6x) –14x – 12x2 Multiply to check your answer. The product is the original polynomial. Caution! When you factor out –1 as the first step, be sure to include it in all the other steps as well. Example 17: Factoring by Using the GCF Factor the polynomial. Check your answer. 3x3 + 2x2 – 10 3x3 = 3 2x2 = 10 = x x x Find the GCF. 2 xx 25 3x3 + 2x2 – 10 There are no common factors other than 1. The polynomial cannot be factored further. Example 18 Factor the polynomial. Check your answer. 5b + 9b3 5b = 5 b 9b = 3 3 b b b b 5(b) + 9b2(b) b(5 + 9b2) Check b(5 + 9b2) 5b + 9b3 Find the GCF. The GCF of 5b and 9b3 is b. Write terms as products using the GCF as a factor. Use the Distributive Property to factor out the GCF. Multiply to check your answer. The product is the original polynomial. Example 19 Factor the polynomial. Check your answer. 9d2 – 82 9d2 = 3 3 d d 82 = 9d2 – 82 Find the GCF. 222222 There are no common factors other than 1. The polynomial cannot be factored further. Example 20 Factor the polynomial. Check your answer. –18y3 – 7y2 – 1(18y3 + 7y2) Both coefficients are negative. Factor out –1. 18y3 = 2 3 3 y y y Find the GCF. 7y2 = 7 yy y y = y2 The GCF of 18y3 and 7y2 is y2. –1[18y(y2) + 7(y2)] –1[y2(18y + 7)] –y2(18y + 7) Write each term as a product using the GCF. Use the Distributive Property to factor out the GCF.. Example 21 Factor each polynomial. Check your answer. 8x4 + 4x3 – 2x2 8x4 = 2 2 2 x x x x 4x3 = 2 2 x x x Find the GCF. 2x2 = 2 xx 2 x x = 2x2 The GCF of 8x4, 4x3 and –2x2 is 2x2. 4x2(2x2) + 2x(2x2) –1(2x2) Write terms as products using the 2x2(4x2 + 2x – 1) Check 2x2(4x2 + 2x – 1) 8x4 + 4x3 – 2x2 GCF as a factor. Use the Distributive Property to factor out the GCF. Multiply to check your answer. The product is the original polynomial. To write expressions for the length and width of a rectangle with area expressed by a polynomial, you need to write the polynomial as a product. You can write a polynomial as a product by factoring it. Example 22 The area of a court for the game squash is 9x2 + 6x m2. Factor this polynomial to find possible expressions for the dimensions of the squash court. A = 9x2 + 6x = 3x(3x) + 2(3x) = 3x(3x + 2) The GCF of 9x2 and 6x is 3x. Write each term as a product using the GCF as a factor. Use the Distributive Property to factor out the GCF. Possible expressions for the dimensions of the squash court are 3x m and (3x + 2) m. Example 23 The area of the solar panel on another calculator is (2x2 + 4x) cm2. Factor this polynomial to find possible expressions for the dimensions of the solar panel. A = 2x2 + 4x = x(2x) + 2(2x) = 2x(x + 2) The GCF of 2x2 and 4x is 2x. Write each term as a product using the GCF as a factor. Use the Distributive Property to factor out the GCF. Possible expressions for the dimensions of the solar panel are 2x cm, and (x + 2) cm. Sometimes the GCF of terms is a binomial. This GCF is called a common binomial factor. You factor out a common binomial factor the same way you factor out a monomial factor. Example 24: Factoring Out a Common Binomial Factor Factor each expression. A. 5(x + 2) + 3x(x + 2) 5(x + 2) + 3x(x + 2) (x + 2)(5 + 3x) The terms have a common binomial factor of (x + 2). Factor out (x + 2). B. –2b(b2 + 1)+ (b2 + 1) –2b(b2 + 1) + (b2 + 1) The terms have a common binomial factor of (b2 + 1). –2b(b2 + 1) + 1(b2 + 1) (b2 + 1) = 1(b2 + 1) (b2 + 1)(–2b + 1) Factor out (b2 + 1). Example 25: Factoring Out a Common Binomial Factor Factor each expression. C. 4z(z2 – 7) + 9(2z3 + 1) There are no common – 7) + + 1) factors. The expression cannot be factored. 4z(z2 9(2z3 Example 26 Factor each expression. a. 4s(s + 6) – 5(s + 6) 4s(s + 6) – 5(s + 6) (4s – 5)(s + 6) The terms have a common binomial factor of (s + 6). Factor out (s + 6). b. 7x(2x + 3) + (2x + 3) 7x(2x + 3) + (2x + 3) The terms have a common binomial factor of (2x + 3). 7x(2x + 3) + 1(2x + 3) (2x + 1) = 1(2x + 1) (2x + 3)(7x + 1) Factor out (2x + 3). Example 27 Factor each expression. c. 3x(y + 4) – 2y(x + 4) 3x(y + 4) – 2y(x + 4) There are no common factors. The expression cannot be factored. d. 5x(5x – 2) – 2(5x – 2) 5x(5x – 2) – 2(5x – 2) (5x – 2)(5x – 2) (5x – 2)2 The terms have a common binomial factor of (5x – 2 ). (5x – 2)(5x – 2) = (5x – 2)2 You may be able to factor a polynomial by grouping. When a polynomial has four terms, you can make two groups and factor out the GCF from each group. Example 28: Factoring by Grouping Factor each polynomial by grouping. Check your answer. 6h4 – 4h3 + 12h – 8 (6h4 – 4h3) + (12h – 8) Group terms that have a common number or variable as a factor. 2h3(3h – 2) + 4(3h – 2) Factor out the GCF of each group. 2h3(3h – 2) + 4(3h – 2) (3h – 2) is another common factor. (3h – 2)(2h3 + 4) Factor out (3h – 2). Example 28 Continued Factor each polynomial by grouping. Check your answer. Check (3h – 2)(2h3 + 4) Multiply to check your solution. 3h(2h3) + 3h(4) – 2(2h3) – 2(4) 6h4 + 12h – 4h3 – 8 6h4 – 4h3 + 12h – 8 The product is the original polynomial. Example 29: Factoring by Grouping Factor each polynomial by grouping. Check your answer. 5y4 – 15y3 + y2 – 3y (5y4 – 15y3) + (y2 – 3y) Group terms. 5y3(y – 3) + y(y – 3) Factor out the GCF of each group. 5y3(y – 3) + y(y – 3) (y – 3) is a common factor. (y – 3)(5y3 + y) Factor out (y – 3). Example 29 Continued Factor each polynomial by grouping. Check your answer. 5y4 – 15y3 + y2 – 3y Check (y – 3)(5y3 + y) y(5y3) + y(y) – 3(5y3) – 3(y) Multiply to check your solution. 5y4 + y2 – 15y3 – 3y 5y4 – 15y3 + y2 – 3y The product is the original polynomial. Example 30 Factor each polynomial by grouping. Check your answer. 6b3 + 8b2 + 9b + 12 (6b3 + 8b2) + (9b + 12) Group terms. 2b2(3b + 4) + 3(3b + 4) 2b2(3b + 4) + 3(3b + 4) Factor out the GCF of each group. (3b + 4) is a common factor. (3b + 4)(2b2 + 3) Factor out (3b + 4). Example 30 Continued Factor each polynomial by grouping. Check your answer. 6b3 + 8b2 + 9b + 12 Check (3b + 4)(2b2 + 3) Multiply to check your solution. 3b(2b2) + 3b(3)+ (4)(2b2) + (4)(3) 6b3 + 9b+ 8b2 + 12 6b3 + 8b2 + 9b + 12 The product is the original polynomial. Example 31 Factor each polynomial by grouping. Check your answer. 4r3 + 24r + r2 + 6 (4r3 + 24r) + (r2 + 6) Group terms. 4r(r2 + 6) + 1(r2 + 6) Factor out the GCF of each group. (r2 + 6) is a common factor. 4r(r2 + 6) + 1(r2 + 6) (r2 + 6)(4r + 1) Factor out (r2 + 6). Example 32 Continued Factor each polynomial by grouping. Check your answer. Check (4r + 1)(r2 + 6) 4r(r2) + 4r(6) +1(r2) + 1(6) Multiply to check your solution. 4r3 + 24r +r2 + 6 4r3 + 24r + r2 + 6 The product is the original polynomial. Helpful Hint If two quantities are opposites, their sum is 0. (5 – x) + (x – 5) 5–x+x–5 –x+x+5–5 0+0 0 Recognizing opposite binomials can help you factor polynomials. The binomials (5 – x) and (x – 5) are opposites. Notice (5 – x) can be written as –1(x – 5). –1(x – 5) = (–1)(x) + (–1)(–5) Distributive Property. = –x + 5 Simplify. =5–x Commutative Property of Addition. So, (5 – x) = –1(x – 5) Example 33: Factoring with Opposites Factor 2x3 – 12x2 + 18 – 3x 2x3 – 12x2 + 18 – 3x (2x3 – 12x2) + (18 – 3x) 2x2(x – 6) + 3(6 – x) 2x2(x – 6) + 3(–1)(x – 6) 2x2(x – 6) – 3(x – 6) (x – 6)(2x2 – 3) Group terms. Factor out the GCF of each group. Write (6 – x) as –1(x – 6). Simplify. (x – 6) is a common factor. Factor out (x – 6). Example 34 Factor each polynomial. Check your answer. 15x2 – 10x3 + 8x – 12 (15x2 – 10x3) + (8x – 12) 5x2(3 – 2x) + 4(2x – 3) Group terms. Factor out the GCF of each group. 5x2(3 – 2x) + 4(–1)(3 – 2x) Write (2x – 3) as –1(3 – 2x). 5x2(3 – 2x) – 4(3 – 2x) (3 – 2x)(5x2 – 4) Simplify. (3 – 2x) is a common factor. Factor out (3 – 2x). Example 35 Factor each polynomial. Check your answer. 8y – 8 – x + xy (8y – 8) + (–x + xy) Group terms. 8(y – 1)+ (x)(–1 + y) Factor out the GCF of each group. 8(y – 1)+ (x)(y – 1) (y – 1) is a common factor. Factor out (y – 1) . (y – 1)(8 + x) Lesson Quiz: Part I Factor each polynomial. Check your answer. 1. 16x + 20x3 4x(4 + 5x2) 2. 4m4 – 12m2 + 8m 4m(m3 – 3m + 2) Factor each expression. 3. 7k(k – 3) + 4(k – 3) 4. 3y(2y + 3) – 5(2y + 3) (k – 3)(7k + 4) (2y + 3)(3y – 5) Lesson Quiz: Part II Factor each polynomial by grouping. Check your answer. 5. 2x3 + x2 – 6x – 3 (2x + 1)(x2 – 3) 6. 7p4 – 2p3 + 63p – 18 (7p – 2)(p3 + 9) 7. A rocket is fired vertically into the air at 40 m/s. The expression –5t2 + 40t + 20 gives the rocket’s height after t seconds. Factor this expression. –5(t2 – 8t – 4) Warm Up Simplify. 1. 2(w + 1) 2w + 2 2. 3x(x2 – 4) 3x3 – 12x Find the GCF of each pair of monomials. 3. 4h2 and 6h 2h 4. 13p and 26p5 13p