What is prime factorization? Maybe use this number as an example? -117 So final answer is: -1 3 39 3 13 -1 x 32 x 13 GCF – Greatest Common Factor Find the GCF of each set of monomials. 54, 63, 180 9 27a2b & 15ab2c 3ab 8g2h2, 20gh, 36g2h3 4gh Relatively Prime • Define relatively prime, then give an example. If two or more integers or monomials have a GCF of 1, then they are said to be relatively prime. Example: 21m and 25b Factor completely: • 140x3 y2 z -48cd2 2257xxxyyz -1 2 2 2 2 3 c d d 55p2 – 11p4 + 44p5 11p2(5 – p2 + 4p3) Factor completely: 12ax + 3xz + 4ay + yz Since all terms do not have a common factor, use grouping: (12ax + 3xz) + (4ay + yz) 3x (4a + z) + y (4a + z) (3x + y) (4a + z) Factoring Trinomials 2 ax + bx + c Remember to do and check each step: 1) Can the equation be simplified? 2) Is there a GCF? (then take it (factor it) out!) 3) Is it a special pattern: a2 – b2, a2 – 2ab + b2, a2 + 2ab + b2 look for perfect squares!!! 4) No special pattern, then factor! (Use grouping, ac method, illegal or diamond factoring if necessary) a2 – b2 = (a + b)(a – b) a2 – 2ab + b2 = (a – b)2 a2 + 2ab + b2 = (a + b)2 Examples 4x2 + 16 4(x2 + 4) 1) Can it be simplified? NO! 2. Is there a GCF? YES … so factor if out You’re 3. Is it a special pattern? NO! 4. Can it be factored any further? done! Another Example 4x2 – 16 4(x2 – 4) 3.if out YES Is–+itit’s a special the difference 4(x 2)(x – pattern? 2) of squares 2. Is there … factor GCF? 1)YES Can it so besoasimplified? 4. Can it be factored any further? Ta da … you’re done! Did you notice the similarity and the differences between the last 2 problems? Trinomial Examples x2 + 7x + 12 (x + 4)(x + 3) 1) Can it be simplified? You’re done! 2. Is there a GCF? 3. Is it a special pattern? 4. Factor … what are the factors of the last term that add up to the middle term? Trinomial Examples #2 x2 + 3x – 10 (x + 5)(x – 2) 1) Can it be simplified? 2. Is there a GCF? You’re done! 3. Is it a special pattern? 4. Factor … what are the factors of the last term that add up to the middle term? Trinomial Examples #3 2x2 – 11x + 15 (2x – 5)(x – 3) CAREFUL – there’s a number in 1) Can it be simplified? 2! You’re done! front of the x 2. Is there a GCF? I’ll wait while you work it out ….. 3. Is it a special pattern? 4. Factor … use the method of YOUR choice! Trinomial Examples #4 4x2 – 18x – 10 2(2x2 – 9x – 5) 1) Can it be–simplified? CAREFUL there’s a number in 2. Is there fronta GCF? of the x2! 3. Is it a special pattern? 2(x – 5)(2x + 1) I’ll wait while you work it out ….. 4. Factor … use the technique of YOUR choice! You’re done! Difference of Squares a2 – b2 (a + b)(a – b) Example: 4x2 – 25 2x 2x 5 5 (2x + 5)(2x – 5) What would you do? 2 2 48a b 2 6x y – – 12ab 2 21y w +24xw xy – 2xz + 5y – 10z What would you do? – 10a + 21 2 3n – 11n + 6 2 9x – 25 2 x – 6x – 27 = 0 2 a