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Factor: 7r 2 + 8r + 1 Factor the first term: 7r 2 = (7r) (r) 7rr Write them side by side with space The last term is positive and the middle term is positive The last term should have two positive factors +1 = (+1) (+ 1) Write these below 7r and r +1 4.3 More on Factoring Trinomials

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7rr +1 +1 (7r) = 7 r +1(r) = r Adding gives: 7r + r = 8r Pick the factors vertically The first factor is (7r + 1) The second factor is (r + 1) 7r 2 + 8r + 1 = (7r + 1) (r + 1) Multiply diagonally: 7r 2 + 8r + 1

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Factor: 20x 2 + 11x – 3 5x4x 20x 2 can be factored as: 20x 2 = (5x) (4x) -3 = (+3) (-1) Write +3 below 4x and –1 below 5x +3 Write this side by side You need to do this by trial and error

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20x 2 + 11x – 3 5x4x +3 (+3) (5x) = 15x -1 (4x) = -4x Adding gives: 15x – 4x = 11x Pick the factors vertically The first factor is (5x – 1) The second factor is (4x + 3) Multiply diagonally: 20x 2 + 11x – 3 = (5x – 1) (4x + 3)

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Factor: 48b 2 –74b – 10 2 is a common factor 8b3b 48b 2 –74b – 10 = 2 (24b 2 –37b – 5 ) 24b 2 = (8b) (3b) + 1– 5 We now factor inside the parenthesis. – 5 = (– 5)(+1) Write –5 below 3b and +1 below 8b

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2 (24b 2 –37b – 5) 8b3b + 1– 5 (+1) (3b) = 3b – 5 (8b) = – 40b Adding gives: – 40b + 3b = -37b The first factor is (8b + 1) The second factor is (3b – 5) Multiply diagonally: 48b 2 –74b – 10 = 2(8b + 1)(3b – 5)

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Factor: 24a 4 + 10a 3 – 4a 2 2a 2 is a common factor 4a3a 24a 4 + 10a 3 – 4a 2 = 2a 2 (12a 2 + 5a – 2 ) 12a 2 = (4a) (3a) +2 We now factor inside the parenthesis. – 2 = (+2)(-1) Write +2 below 3a and -1 below 4a

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2a 2 (12a 2 + 5a – 2 ) 4a3a +2 (-1) (3a) = -3a +2 (4a) = + 8a Adding gives: 8a – 3a = 5a The first factor is (4a – 1) The second factor is (3a + 2) Multiply diagonally: 24a 4 + 10a 3 – 4a 2 = 2a 2 (4a – 1)(3a + 2)

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Factor: 18 + 65x + 7x 2 92 18 = (9) (2) 7x 2 = (+7x) (+x) Write +7x below 2 and +x below 9 +x+7x Write this side by side You need to do this by trial and error

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18 + 65x + 7x 2 92 +x+7x (+7x) (9) = 63x (x) (2) = 2x Adding gives: 63x + 2x = 65x Pick the factors vertically The first factor is (9 + x) The second factor is (2 + 7x) Multiply diagonally: 18 + 65x + 7x 2 = (9 +x) (2 + 7x) =(x + 9)(7x + 2)

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Factor: -18k 3 – 48k 2 + 66k -6k is a common factor 3kk -18k 3 – 48k 2 + 66k = -6k (3k 2 + 8k – 11 ) 3k 2 = (3k) (k) +11 We now factor inside the parenthesis. – 11 = (+11)(-1) Write -1 below k and +11 below 3k

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3kk +11 (11) (k) = 11k -1 (3k) = -3k Adding gives: -3k + 11k = 8k The first factor is (3k + 11) The second factor is (k –1) Multiply diagonally: -18k 3 – 48k 2 + 66k = -6k(3k + 11)(k – 1) -6k (3k 2 + 8k – 11 )

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Factor: 12k 3 q 4 – 4k 2 q 5 – kq 6 kq 4 is the common factor 6k2k 12k 3 q 4 – 4k 2 q 5 – kq 6 = kq 4 (12k 2 – 4kq – q 2 ) 12k 2 = (6k) (2k) +q-q We now factor inside the parenthesis. – q 2 = (+q)(-q) Write -q below 2k and +q below 6k

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6k2k +q-q (q) (2k) = 2kq -q (6k) = -6kq Adding gives: -6kq + 2kq = -4kq The first factor is (6k + q) The second factor is (2k –q) Multiply diagonally: 12k 3 q 4 – 4k 2 q 5 – kq 6 = kq 4 (6k + q)(2k – q) kq 4 (12k 2 – 4kq – q 2 )

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Factor: 14a 2 b 3 +15ab 3 – 9b 3 b 3 is the common factor 7a2a 14a 2 b 3 +15ab 3 – 9b 3 = b 3 (14a 2 + 15a – 9 ) 14a 2 = (7a) (2a) -3+3 We now factor inside the parenthesis. – 9 = (+3)(-3) Write +3 below 2a and -3 below 7a

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b 3 (14a 2 + 15a – 9 ) 7a2a -3 +3 (-3) (2a) = - 6a 3 (7a) = 21a Adding gives: 21a – 6a = 15a The first factor is (7a – 3) The second factor is (2a + 3) Multiply diagonally: 14a 2 b 3 +15ab 3 – 9b 3 = b 3 (7a – 3)(2a + 3)

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