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5 Minute Warm-Up

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  1. 5 Minute Warm-Up Directions: Simplify each problem. Write the answer in standard form. • (4x + 7)(– 2x) • -4x2(3x2 + 2x – 6) • (x + 6)(x + 9) • (-4y + 5)(-7 – 3y) • (-8x3 + x – 9x2 + 2) + (8x2 – 2x + 4) 6. (6x2 – x + 3) – (-2x + x2 – 7)

  2. Warm Up 1. 50, 62. 105, 7 3. List the factors of 28. Tell whether each number is prime or composite. If the number is composite, write it as the product of two numbers. Tell whether the second number is a factor of the first number no yes ±1, ±2, ±4, ±7, ±14, ±28 prime composite; 49  2 4. 11 5. 98

  3. Objectives Write the prime factorization of numbers. Find the GCF of monomials.

  4. 1 12 3 4 1 4 3     2 6 2 2 3    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

  5. 1 12 3 4 1 4 3     2 6 2 2 3    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

  6. Remember! A prime number has exactly two factors, itself and 1. The number 1 is not prime because it only has one factor.

  7. 98 2 98 49  7 2 49 7 7  7 7 1 98 = 2 7 7 98 = 2 7 7     Example 1: Writing Prime Factorizations Write the prime factorization of 98. Method 1 Factor tree Method 2 Ladder diagram Choose any two factors of 98 to begin. Keep finding factors until each branch ends in a prime factor. Choose a prime factor of 98 to begin. Keep dividing by prime factors until the quotient is 1. The prime factorization of 98 is 2  7  7 or 2 72.

  8. 40 33 11  2 20 3  2 10  2 5 Check It Out! Example 1 Write the prime factorization of each number. a. 40 b. 33 40 = 23 5 33 = 3  11 The prime factorization of 40 is 2  2  2  5 or 23 5. The prime factorization of 33 is 3  11.

  9. 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.

  10. Example 2A: 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.

  11. Example 2B: Finding the GCF of Numbers Find the GCF of each pair of numbers. 26 and 52 Method 2 Prime factorization. Write the prime factorization of each number. 26 = 2  13 52 = 2  2  13 Align the common factors. 2  13 = 26 The GCF of 26 and 52 is 26.

  12. 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.

  13. Example 3A: Finding the GCF of Monomials Find the GCF of each pair of monomials. 15x3 and 9x2 Write the prime factorization of each coefficient and write powers as products. 15x3 = 3  5  x  x  x Align the common factors. 9x2 = 3  3 x  x 3 x  x = 3x2 Find the product of the common factors. The GCF of 3x3 and 6x2 is 3x2.

  14. Example 3B: Finding the GCF of Monomials Find the GCF of each pair of monomials. 8x2 and 7y3 The GCF 8x2 and 7y is 1.

  15. Helpful Hint If two terms contain the same variable raised to different powers, the GCF will contain that variable raised to the lower power.

  16. Check It Out! Example 3a Find the GCF of each pair of monomials. 18g2 and 27g3 Write the prime factorization of each coefficient and write powers as products. 18g2 = 2  3  3  g  g Align the common factors. 27g3 = 3  3  3 g  g  g 3  3 g  g Find the product of the common factors. The GCF of 18g2 and 27g3 is 9g2.

  17. Check It Out! Example 3b Find the GCF of each pair of monomials. Write the prime factorization of each coefficient and write powers as products. 16a6 and 9b 16a6 = 2  2  2  2 a  a  a  a  a  a 9b = 3  3 b Align the common factors. The GCF of 16a6 and 7b is 1. There are no common factors other than 1.

  18. Check It Out! Example 4 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.

  19. Check It Out! Example 4 Continued Find the common factors of 36 and 48. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 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.

  20. 36 pop CDs = 3 shelves 12 CDs per shelf 48 country CDs = 4 shelves 12 CDs per shelf When the greatest possible number of CD types are on each shelf, there are 7 shelves in total.

  21. 5 Minute Warm-Up Write the prime factorization of each number. 1. 50 2. 84 Find the GCF of each pair monomial. 3. 18 and 75 4. 20 and 36 5. 12x and 28x36. 27x2 and 45x3y2 7. Cindi is planting a rectangular flower bed with 40 orange flower and 28 yellow flowers. She wants to plant them so that each row will have the same number of plants but of only one color. How many rows will Cindi need if she puts the greatest possible number of plants in each row?

  22. Objective Factor polynomials by using the greatest common factor.

  23. 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.

  24. Example 1A: Factoring by Using the GCF Factor each polynomial. Check your answer. 2x2– 4 2x2 = 2 x  x Find the GCF. 4 = 2  2 The GCF of 2x2 and 4 is 2. 2 Write terms as products using the GCF as a factor. 2x2–(2 2) 2(x2– 2) Use the Distributive Property to factor out the GCF. Multiply to check your answer. Check 2(x2– 2) The product is the original polynomial.  2x2– 4

  25. Writing Math Aligning common factors can help you find the greatest common factor of two or more terms.

  26. Example 1B: Factoring by Using the GCF Factor each polynomial. Check your answer. 8x3– 4x2– 16x 8x3 = 2  2  2  x  x  x Find the GCF. 4x2 = 2  2  x  x 16x = 2  2  2  2 x The GCF of 8x3, 4x2, and 16x is 4x. Write terms as products using the GCF as a factor. 2  2 x = 4x 2x2(4x)–x(4x) – 4(4x) Use the Distributive Property to factor out the GCF. 4x(2x2–x– 4) Check 4x(2x2–x– 4) Multiply to check your answer. The product is the original polynomials. 8x3– 4x2– 16x 

  27. Example 1C: Factoring by Using the GCF Factor each polynomial. Check your answer. –14x– 12x2 Both coefficients are negative. Factor out –1. – 1(14x + 12x2) 14x = 2  7  x 12x2 = 2  2  3  x  x Find the GCF. The GCF of 14x and 12x2 is 2x. 2 x = 2x –1[7(2x) + 6x(2x)] Write each term as a product using the GCF. –1[2x(7 + 6x)] Use the Distributive Property to factor out the GCF. –2x(7 + 6x)

  28. Caution! When you factor out –1 as the first step, be sure to include it in all the other steps as well.

  29. Example 1D: Factoring by Using the GCF Factor each polynomial. Check your answer. 3x3 + 2x2– 10 3x3 = 3 x x x Find the GCF. 2x2 = 2 x x 10 = 2 5 There are no common factors other than 1. 3x3 + 2x2– 10 The polynomial cannot be factored further.

  30. Check b(5 + 9b2) Check It Out! Example 1a Factor each polynomial. Check your answer. 5b + 9b3 5b = 5 b Find the GCF. 9b = 3 3 b b b The GCF of 5b and 9b3 is b. b Write terms as products using the GCF as a factor. 5(b) + 9b2(b) Use the Distributive Property to factor out the GCF. b(5 + 9b2) Multiply to check your answer. The product is the original polynomial. 5b + 9b3 

  31. Check It Out! Example 1b Factor each polynomial. Check your answer. 9d2– 82 Find the GCF. 9d2 = 3 3 d d 82 = 2  2  2  2  2  2 There are no common factors other than 1. 9d2– 82 The polynomial cannot be factored further.

  32. Check It Out! Example 1c Factor each 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 y  y y y =y2 The GCF of 18y3 and 7y2 is y2. Write each term as a product using the GCF. –1[18y(y2) + 7(y2)] –1[y2(18y + 7)] Use the Distributive Property to factor out the GCF.. –y2(18y + 7)

  33. Check It Out! Example 1d 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 x x 2 x x = 2x2 The GCF of 8x4, 4x3 and –2x2 is 2x2. Write terms as products using the GCF as a factor. 4x2(2x2) + 2x(2x2)–1(2x2) 2x2(4x2 + 2x– 1) Use the Distributive Property to factor out the GCF. Check 2x2(4x2 + 2x– 1) Multiply to check your answer. 8x4 + 4x3– 2x2 The product is the original polynomial.

  34. 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.

  35. Example 2: Application 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 The GCF of 9x2 and 6x is 3x. Write each term as a product using the GCF as a factor. = 3x(3x) + 2(3x) = 3x(3x + 2) 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.

  36. 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.

  37. Example 3: Factoring Out a Common Binomial Factor Factor each expression. A. 5(x + 2) + 3x(x + 2) The terms have a common binomial factor of (x + 2). 5(x + 2) + 3x(x + 2) (x + 2)(5 + 3x) Factor out (x + 2). B. –2b(b2 + 1)+ (b2 + 1) The terms have a common binomial factor of (b2 + 1). –2b(b2 + 1) + (b2 + 1) –2b(b2 + 1) + 1(b2 + 1) (b2 + 1) = 1(b2 + 1) (b2 + 1)(–2b + 1) Factor out (b2 + 1).

  38. Check It Out! Example 3 Factor each expression. a. 4s(s + 6) – 5(s + 6) The terms have a common binomial factor of (s + 6). 4s(s + 6)– 5(s + 6) (4s – 5)(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).

  39. 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.

  40. Example 4A: Factoring by Grouping Factor each polynomial by grouping. Check your answer. 6h4– 4h3 + 12h– 8 Group terms that have a common number or variable as a factor. (6h4– 4h3) + (12h– 8) 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).

  41. 6h4– 4h3 + 12h– 8 Example 4A 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 The product is the original polynomial.

  42. Example 4B: Factoring by Grouping Factor each polynomial by grouping. Check your answer. 5y4– 15y3 + y2– 3y (5y4– 15y3) + (y2– 3y) Group terms. Factor out the GCF of each group. 5y3(y – 3) + y(y– 3) (y – 3) is a common factor. 5y3(y– 3) + y(y– 3) (y– 3)(5y3 + y) Factor out (y – 3).

  43. Check It Out! Example 4a Factor each polynomial by grouping. Check your answer. 6b3 + 8b2 + 9b + 12 (6b3 + 8b2) + (9b + 12) Group terms. Factor out the GCF of each group. 2b2(3b + 4) + 3(3b + 4) (3b + 4) is a common factor. 2b2(3b + 4) + 3(3b + 4) Factor out (3b + 4). (3b + 4)(2b2 + 3)

  44. Check It Out! Example 4b 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. 4r(r2 + 6) + 1(r2 + 6) (r2 + 6) is a common factor. (r2 + 6)(4r + 1) Factor out (r2 + 6).

  45. 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

  46. 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. Simplify. = –x + 5 = 5 –x Commutative Property of Addition. So, (5 –x) = –1(x– 5)

  47. Example 5: Factoring with Opposites Factor 2x3– 12x2 + 18 – 3x 2x3– 12x2 + 18 – 3x (2x3– 12x2) + (18 – 3x) Group terms. 2x2(x– 6) + 3(6 –x) Factor out the GCF of each group. 2x2(x– 6) + 3(–1)(x– 6) Write (6 – x) as –1(x – 6). 2x2(x– 6) – 3(x– 6) Simplify. (x – 6) is a common factor. (x –6)(2x2– 3) Factor out (x – 6).

  48. Check It Out! Example 5a Factor each polynomial. Check your answer. 15x2– 10x3 + 8x– 12 (15x2– 10x3) + (8x– 12) Group terms. Factor out the GCF of each group. 5x2(3 – 2x) + 4(2x– 3) 5x2(3 – 2x) + 4(–1)(3 – 2x) Write (2x – 3) as –1(3 – 2x). Simplify. (3 – 2x) is a common factor. 5x2(3 – 2x) – 4(3 – 2x) (3 – 2x)(5x2– 4) Factor out (3 – 2x).

  49. 5 Minute Warm-Up Factor each polynomial. 1. 16x + 20x3 2. 4m4 – 12m2 + 8m 3. 7k(k – 3) + 4(k – 3) 4. 3y(2y + 3) – 5(2y + 3) 5. 2x3 + x2 – 6x – 3 6. 7p4 – 2p3 + 63p – 18

  50. Objective Factor quadratic trinomials of the form x2 + bx + c.