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Exercises: (1) Find the gcf of the expressions 12x3 and 18x4. Exercises: (2) Factor out the gcf: (a) 5 x3 10 x 2 15 x (b) 4 x 2 y 2 8xy 2 12 xy (3) Factor by grouping: (a) rx + 2rs + 2x + 4s. (b) 3xy + 6y -2x - 4. (4) Factor: (a) x2 - 9, (b) 7x2 - 700, (c) 2x4 - 2. (5) Factor the polynomial: (a) x 2 8 x 7. (b) x 2 2 x 15 . (c) x 2 x 20. (d) x 2 8 x 12. (e) 7 x 2 14 xy 21y . (6) Factor the polynomial: (a) 6x2 – x – 15 (b) 5x2 – 7x – 6 (c) 8x2 + 22x – 21 (7) Factor the polynomial: (a) x3 1000 (b) 27 x3 64 (c) 5 x 3 5 (8) Factor the polynomial: (a) x 2 10 x 25. (b) 4 x 2 36 x. (c) 4 x3 36 x. (d) 3 x 2 11x 4. (e) x3 216. (f) 4 x 2 8 x 7 x 14. (g) 6 x 2 24 (h) 1000 x3 27 (i) x 2 11x 30. (j) 2 x 2 7 x 15. (9) Solve the polynomial equation: (a) x 2 25 0. (b) x 2 x 6. (c) 2 x3 10 x 2 28 x. Solutions: (1) the gcf of the coefficients 12 and 13 is 6. The lowest power of x is x3. So the GCF = 6x3. (2) (a) 5 x3 10 x 2 15 x = 5x(x2 + 2x - 3) 5x3 10x2 5x2 5x2 (the gcf = 5x). -15x 5x2 (b) 4 x 2 y 2 8xy 2 12 xy 4xy(xy - 2y + 3). (gcf = 4xy) 4x2y2 -8xy2 12xy 4xy 4xy 4xy (3) (a) rx + 2rs + 2x + 4s. gcf = r gcf = 2 r ( x 2s) 2( x 2s) (r 2)( x 2 s ) (b) 3xy + 6y -2x - 4 gcf = 3y gcf = -2 3 y ( x 2) 2( x 2) (3 y 2)( x 2) . (4) (a) x2 - 9, (x)2 - 32 (pair the factors and find the gcf) (factor out the gcf from the pairs) (combine like terms on top of (x + 2s) by adding the coefficients r and 2) (pair off the terms and find the gcf:) (Note that I needed to make the signs match by using gcf = -2 in the second pair. Now pull out the gcf's:) (Now combine like terms on (x + 2) ) (DOS ) (split it up with the formula) = (x + 3)(x - 3). (b) 7x2 - 700 = 7(x2 - 100) = (x)2 - 102 7(x - 10)(x + 10). (gcf = 7, factor it out:) (DOS in pharentheses). (c) 2x4 - 2 = 2(x4 - 1) = (x2)2 - 12, 2( x 2 1)( x 2 1) = 2( x 1)( x 1)( x 2 1) (gcf = 2, factor it out:) (rewrite x4 - DOS in pharentheses:) (split it up using formula:) (x2 - 1 is a DOS - split it up again) (the complete factorization). (5) x 2 8 x 7. 1 7 = (x + 1)(x + 7). (need two numbers that multiply to 7 add to 8 - there ain't many choices since 7 is prime - only 1 and 7 are factors) (1 + 7 = 8, so:) (b) x 2 2 x 15 (need two numbers which * to -15, add to -2 list the factors of -15. The larger ones will have to be 1 -15 negative.) 3 -5 (3 + -5 = -2, so: ) = (x + 3)(x - 5). (c) x 2 x 20. (need two numbers which * to -20, add to +1 - the larger factor has to be positive this time, ) -1 20 -2 10 -4 5 (-4 + 5 = 1) = (x - 4)(x + 5). (d) x 2 8 x 12. (need two numbers which * to 12, add to -8 - both factors will have to be negative to add to a negative, * to a positive) -1 -12 -2 -6 (-2 + -6 = -8) -3 -4 = (x - 2)(x - 6). (e) 7 x 2 14 xy 21y = 7(x2 + 2xy - 3y) = -1 3 7(x - y)(x + 3y). (first, factor out the gcf = 7) (find the two numbers which * to -3, + to 2) (add a y to the -1 and 3:) (6) (a) 6x2 – x – 15 (6)(-15) = -90 (multiply the lead and constant terms) 1 -90 2 -45 3 -30 5 -18 6 -15 9 -10 = 6x2 + 9x - 10x - 15 (9 + -10 = -1 - use these to split up -x:) (pair the terms and get the gcf's) gcf = 3x gcf = -5 = 3x(2x + 3) - 5(2x + 3) = (3x - 5)(2x + 3). (picked - 5 to make signs match) (factor out gcf's, combine terms: ) (voila. finito. ) (b) 5x2 – 7x – 6 5(-6) = -30, (Multiply the lead and constant terms:) 1 -30 2 -15 3 -10 2 5x - 10x + 3x - 6 (3 + -10 = -7 - use them to split up -7x) (pair off, gcf's:) gcf = 5x gcf = 3 5x(x - 2) + 3(x - 2) = (5x + 3)(x - 2). (factor gcf's, combine terms: ) (c) 8x2 + 22x – 21 (8)(-21) = -168 (multiply constant and lead term:) -1 168 -2 84 -3 56 -4 42 -6 28 (-6 + 28 = 22 : use them to break up 22x) 8x2 + 28x - 6x - 21 = (pair off, find gcf's, make signs match) gcf = 4x gcf = -3 4x(2x + 7) - 3(2x + 7) = (4x - 3)(2x + 7). (factor out gcf's) (combine like terms:) (rock-action) (7) (a) x3 1000 (This is a DOC) 3 3 (x) - 10 (use formula with a = x, b = 10). (x - 10)(x2 + 10x + 100). a - b a2 ab b2 (b) 27 x3 64 (This is a SOC) (3x)3 + (4)3 = (use formula, with a = 3x, b = 4). 2 (3x + 4)(9x - 12x + 16) (answer) (3x)2 (3x)(4) (4)2 (c) 5 x 3 5 5(x3 - 1) 5( x 1)( x2 x 1). (8)(a) x 2 10 x 25. -1 -25 -5 -5 = (x - 5)(x - 5). (b) 4 x 36 x. 2 = 4 x( x 9) (c) 4 x3 36 x. = 4 x( x 2 9) = 4 x( x 3)( x 3). (gcf = 5, take it out: ) (DOC inside pharentheses, w/ a = x, b = 1) (yeah) (no gcf, trinomial, lead coeff = 1) (need two numbers which * to 25, + to -10: -5 + -5 = -10 ) 4 x2 36 x (gcf = 4x, factor it out: x, 9 .) 4x 4x no more factoring can be done) (gcf = 4x, factor it out:) (DOS in pharentheses (x)2 - 32) (complete factorization) (d) 3 x 2 11x 4. 3*4 = 12 (no gcf, trinomial, lead coeff = 3, grouping) 1 -12 = 3x x 12 x 4 (use 1 and -12 to break up 11x) (pair off, gcf's) 2 gcf = x, gcf = -4 (signs have to match) = x(3x 1) 4(3 x 1) (combine terms) = ( x 4)(3 x 1). (e) x3 + 216 = (x)3 + (6)3 ( x 3)( x 2 3x 9). a + b a2 -ab + b2 (no gcf, two terms, 6 is the cube of 216, so SOC) ( x plays the role of a, 6 plays the role of b) (f) 4 x 2 8 x 7 x 14. (no gcf, four terms - grouping, pair off, gcf's) gcf = 4x gcf = 7 4 x( x 2) 7( x 2) (4 x 7)( x 2). (factor out gcf's) (combine terms.) (g) 6 x 2 24 (gcf = 6, factor it out:) 6( x 2 4) 6( x 2)( x 2). (two terms, DOS in pharentheses (x)2 - 22 ) (split it up - complete factorization) (h) 1000 x3 27 (no gcf, two terms, note the cube x3, and 1000 = 103, 27 = 33) 3 3 (10x) - 3 (DOC, w/ 10x playing a, 3 playing b) 2 (10 x 3)(100 x 30 x 9) . (10x)2 3*10x 32 (i) x 2 11x 30. (no gcf, trinomial, lead coefficient = 1) 1 30 2 15 3 10 5 6 = (x + 5)(x + 6). (j) 2 x 2 7 x 15. (2)(-15)=-30 ( 5 + 6 =11) (no gcf, trinomial, lead coefficient 1, grouping:) -1 30 -2 15 -3 10 (-3 + 10 = 7 - use them to break up 7x) 2 = 2 x 3x 10 x 15 (pair off, gcf's) gcf = x gcf = 5 (pull out gcf's) = x(2 x 3) 5(2 x 3) (combine terms:) ( x 5)(2 x 3). (9) (a) x 2 25 0. (x + 5)(x - 5) = 0, x + 5 = 0, x - 5 = 0, -5 -5 +5 +5 x = 5, x = -5. x = {5} (b) x 2 x 6. -6 -6 2 x x 6 0, ( x 3)( x 2) 0, x 3 0, x 2 0. +3 +3 -2 -2 x = 3, x = -2, x = {3, -2} (c) 2 x3 10 x 2 28 x. (LHS = 0, so, factor LHS as a DOS) (set factors = 0,) (give answer in solution set: ) (get LHS = 0) (factor LHS) (set factors = 0) (get LHS = 0) -28x -28x 2 x 10 x 28 x 0 2 x( x 2 5x 14) 0, 3 2 (gcf = 2x) (inside pharentheses - trinomial, lead coefficient = 1) -1 14 -2 7 (-2 + 7 = 5) 2 x( x 7)( x 2) 0, (factors = 0) 2 x 0, x 7 0, x 2 0, x= 0 -7 -7 +2 +2, x = 0, x = -7, x = 7. x = {0,7}