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1. find the prime factorization of a number. 2. find the greatest common factor (GCF) for a set of monomials. SOL: A.2c Designed by Skip Tyler, Varina High School number that can only be divided by only one and itself. A composite number is a number greater than one that is not prime. Prime or composite? 37 prime 51 composite 1. 2. 3. 4. Prime Composite Both Neither 84 = 4 • 21 = 2•2•3•7 = 22 • 3 • 7 2) Find the prime factorization of -210. -210 = -1 • 210 = -1 • 30 • 7 = -1 • 6 • 5 • 7 = -1 • 2 • 3 • 5 • 7 45a2b3 = b = 9•5•a•a•b•b•b 3•3•5•a•a•b•b• = 32 • 5 • a • a • b • b • b Write the variables without exponents. 1. 2. 3. 4. 3 16 344 2234 22223 the largest number that can divide into all of the numbers. 4) Find the GCF of 42 and 60. Write the prime factorization of each number. 42 = 2 • 3 • 7 60 = 2 • 2 • 3 • 5 What prime factors do the numbers have in common? Multiply those numbers. The GCF is 2 • 3 = 6 6 is the largest number that can go into 42 and 60! 40a2b = 2 • 2 • 2 • 5 a • a • b 48ab4 = 2 • 2 • 2 • 2 • 3 • a • b • b • b•b What do they have in common? Multiply the factors together. GCF = 8ab 1. 2. 3. 4. 2 4 8 16 5a Let’s go one step further… 1) FACTOR 25a2 + 15a. Find the GCF and divide each term 25a2 + 15a = 5a( ___ 5a + ___ 3 ) 25a 2 5a 15a 5a Check your answer by distributing. Find the GCF 6x2 Divide each term by the GCF 3 - ___ 2x ) 18x2 - 12x3 = 6x2( ___ 18 x 2 6x2 12 x 3 6x2 Check your answer by distributing. GCF = 28ab Divide each term by the GCF 2 ) a + 2c 28a2b + 56abc2 = 28ab ( ___ ___ 28a 2b 28ab 56abc 2 28ab Check your answer by distributing. 28ab(a + 2c2) 1. 2. 3. 4. x(20 – 24y) 2x(10x – 12y) 4(5x2 – 6xy) 4x(5x – 6y) GCF = 7 Divide each term by the GCF 4a2 + ___ 3b - ____ 5b2c2) 28a2 + 21b - 35b2c2 = 7 ( ___ 28a 7 2 21b 7 35b 2 c 2 7 Check your answer by distributing. 7(4a2 + 3b – 5b2c2) 1. 2. 3. 4. 2y2(8x – 12z + 20) 4y2(4x – 6z + 10) 8y2(2x - 3z + 5) 8xy2z(2 – 3 + 5) Factoring Chart This chart will help you to determine which method of factoring to use. Type Number of Terms 1. GCF 2. Difference of Squares 2 or more 2 Determine the pattern 1 4 9 16 25 36 … = 12 = 22 = 32 = 42 = 52 = 62 These are perfect squares! You should be able to list the first 15 perfect squares in 30 seconds… Perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 Review: Multiply (x – 2)(x + 2) First terms: x2 Outer terms: +2x Inner terms: -2x Last terms: -4 Combine like terms. x2 – 4 Notice the middle terms eliminate each other! x -2 x2 -2x +2 +2x -4 x This is called the difference of squares. Difference of Squares 2 2 a - b = (a - b)(a + b) or 2 2 a - b = (a + b)(a - b) The order does not matter!! 4 Steps for factoring Difference of Squares 1. Are there only 2 terms? 2. Is the first term a perfect square? 3. Is the last term a perfect square? 4. Is there subtraction (difference) in the problem? If all of these are true, you can factor using this method!!! 1. Factor x2 - 25 When factoring, use your factoring table. Do you have a GCF? No Are the Difference of Squares steps true? x2 – 25 Two terms? Yes 1st term a perfect square? Yes 2nd term a perfect square? Yes Subtraction? Yes ( x + 5 )(x - 5 ) Write your answer! 2. Factor 16x2 - 9 When factoring, use your factoring table. Do you have a GCF? No Are the Difference of Squares steps true? 16x2 – 9 Two terms? Yes 1st term a perfect square? Yes 2nd term a perfect square? Yes Subtraction? Yes (4x + 3 )(4x - 3 ) Write your answer! 3. Factor 81a2 – 49b2 When factoring, use your factoring table. Do you have a GCF? No Are the Difference of Squares steps true? 81a2 – 49b2 Two terms? Yes 1st term a perfect square? Yes 2nd term a perfect square? Yes Subtraction? Yes (9a + 7b)(9a - 7b) Write your answer! Factor 1. 2. 3. 4. 2 x (x + y)(x + y) (x – y)(x + y) (x + y)(x – y) (x – y)(x – y) Remember, the order doesn’t matter! – 2 y 4. Factor 2 75x – 12 When factoring, use your factoring table. Do you have a GCF? Yes! GCF = 3 3(25x2 – 4) Are the Difference of Squares steps true? Two terms? Yes 3(25x2 – 4) 1st term a perfect square? Yes 2nd term a perfect square? Yes Subtraction? Yes 3(5x + 2 )(5x - 2 ) Write your answer! Factor 1. 2. 3. 4. 2 18c prime 2(9c2 + 4d2) 2(3c – 2d)(3c + 2d) 2(3c + 2d)(3c + 2d) You cannot factor using difference of squares because there is no subtraction! + 2 8d Factor -64 + 1. 2. 3. 4. prime (2m – 8)(2m + 8) 4(-16 + m2) 4(m – 4)(m + 4) Rewrite the problem as 4m2 – 64 so the subtraction is in the middle! 2 4m Factoring Chart This chart will help you to determine which method of factoring to use. Type Number of Terms 1. GCF 2 or more 2. Difference of Squares 2 3. Grouping 4 1. Factor 12ac + 21ad + 8bc + 14bd Do you have a GCF for all 4 terms? No Group the first 2 terms and the last 2 terms. (12ac + 21ad) + (8bc + 14bd) Find the GCF of each group. 3a (4c + 7d) + 2b(4c + 7d) The parentheses are the same! (3a + 2b)(4c + 7d) 2. Factor rx + 2ry + kx + 2ky Check for a GCF: None You have 4 terms - try factoring by grouping. (rx + 2ry) + (kx + 2ky) Find the GCF of each group. r(x + 2y) + k(x + 2y) The parentheses are the same! (r + k)(x + 2y) 3. Factor 2 2x - 3xz - 2xy + 3yz Check for a GCF: None Factor by grouping. Keep a + between the groups. (2x2 - 3xz) + (- 2xy + 3yz) Find the GCF of each group. x(2x – 3z) + y(- 2x + 3z) The signs are opposite in the parentheses! Keep-change-change! x(2x - 3x) - y(2x - 3z) (x - y)(2x - 3z) 4. Factor 3 16k - 2 2 4k p - 28kp + 3 7p Check for a GCF: None Factor by grouping. Keep a + between the groups. (16k3 - 4k2p2 ) + (-28kp + 7p3) Find the GCF of each group. 4k2(4k - p2) + 7p(-4k + p2) The signs are opposite in the parentheses! Keep-change-change! 4k2(4k - p2) - 7p(4k - p2) (4k2 - 7p)(4k - p2) Objective The student will be able to: factor trinomials with grouping. SOL: A.2c Designed by Skip Tyler, Varina High School Factoring Chart This chart will help you to determine which method of factoring to use. Type Number of Terms 1. GCF 2 or more 2. Diff. Of Squares 2 3. Trinomials 3 Review: (y + 2)(y + 4) y2 +4y +2y +8 First terms: Outer terms: Inner terms: Last terms: Combine like terms. y2 + 6y + 8 y +2 y2 +2y +4 +4y +8 y In this lesson, we will begin with y2 + 6y + 8 as our problem and finish with (y + 2)(y + 4) as our answer. Here we go! 1) Factor y2 + 6y + 8 Use your factoring chart. Do we have a GCF? Nope! Is it a Diff. of Squares problem? No way! 3 terms! Now we will learn Trinomials! You will set up a table with the following information. Product of the first and last coefficients Middle coefficient The goal is to find two factors in the first column that add up to the middle term in the second column. We’ll work it out in the next few slides. 1) Factor 2 y M A + 6y + 8 Create your MAMA table. Product of the first and last coefficients Multiply +8 Add +6 Middle coefficient Here’s your task… What numbers multiply to +8 and add to +6? If you cannot figure it out right away, write the combinations. 1) Factor 2 y + 6y + 8 Place the factors in the table. Multiply +8 Which has a sum of +6? +1, +8 -1, -8 +2, +4 -2, -4 Add +6 +9, NO -9, NO +6, YES!! -6, NO We are going to use these numbers in the next step! 1) Factor y2 + 6y + 8 Multiply +8 Add +6 +2, +4 +6, YES!! Hang with me now! Replace the middle number of the trinomial with our working numbers from the MAMA table y2 + 6y + 8 y2 + 2y + 4y + 8 Now, group the first two terms and the last two terms. We have two groups! (y2 + 2y)(+4y + 8) Almost done! Find the GCF of each group and factor it out. If things are done right, the parentheses y(y + 2) +4(y + 2) should be the same. Factor out the GCF’s. Write them in their own group. (y + 4)(y + 2) Tadaaa! There’s your answer…(y + 4)(y + 2) You can check it by multiplying. Piece of cake, huh? There is a shortcut for some problems too! (I’m not showing you that yet…) M A 2) Factor x2 – 2x – 63 Create your MAMA table. Product of the first and last coefficients Signs need to be different since number is negative. Multiply -63 -63, 1 -1, 63 -21, 3 -3, 21 -9, 7 -7, 9 Add -2 -62 62 -18 18 -2 2 Middle coefficient Replace the middle term with our working numbers. x2 – 2x – 63 x2 – 9x + 7x – 63 Group the terms. (x2 – 9x) (+ 7x – 63) Factor out the GCF x(x – 9) +7(x – 9) The parentheses are the same! Weeedoggie! (x + 7)(x – 9) Here are some hints to help you choose your factors in the MAMA table. 1) When the last term is positive, the factors will have the same sign as the middle term. 2) When the last term is negative, the factors will have different signs. M A 2) Factor 5x2 - 17x + 14 Create your MAMA table. Product of the first and last coefficients Signs need to be the same as the middle sign since the product is positive. Multiply +70 -1, -70 -2, -35 -7, -10 Add -17 -71 -37 -17 Replace the middle term. 5x2 – 7x – 10x + 14 Group the terms. Middle coefficient (5x2 – 7x) (– 10x + 14) Factor out the GCF x(5x – 7) -2(5x – 7) The parentheses are the same! Weeedoggie! (x – 2)(5x – 7) Hopefully, these will continue to get easier the more you do them. Factor 1. 2. 3. 4. (x + 2)(x + 1) (x – 2)(x + 1) (x + 2)(x – 1) (x – 2)(x – 1) 2 x + 3x + 2 Factor 1. 2. 3. 4. (2x + 10)(x + 1) (2x + 5)(x + 2) (2x + 2)(x + 5) (2x + 1)(x + 10) 2 2x + 9x + 10 Factor 1. 2. 3. 4. 2 6y (6y2 – 15y)(+2y – 5) (2y – 1)(3y – 5) (2y + 1)(3y – 5) (2y – 5)(3y + 1) – 13y – 5 2) Factor 2x2 - 14x + 12 Find the GCF! 2(x2 – 7x + 6) Now do the MAMA table! Signs need to be the same as the middle sign since the product is positive. Multiply +6 Add -7 -1, -6 -7 -2, -3 -5 Replace the middle term. 2[x2 – x – 6x + 6] Group the terms. 2[(x2 – x)(– 6x + 6)] Factor out the GCF 2[x(x – 1) -6(x – 1)] The parentheses are the same! Weeedoggie! 2(x – 6)(x – 1) Don’t forget to follow your factoring chart when doing these problems. Always look for a GCF first!!