If I have (15x^2-9x-6 / 3x-3x-6) * (6x^2-6x-12 / 2x^2+2x-4) Can I just simplify it but saying ok 15/3= 5x^2 and -9/-3 is equal to 3x and -6/-6 is 1. Do you see what I mean? Because they can all be cancelled out like that. Is it against the rules??
Simplify the following:
((15 x^2-9 x-6) (6 x^2-6 x-12))/((3 x-3 x-6) (2 x^2+2 x-4))
3 x-3 x = 0:
((15 x^2-9 x-6) (6 x^2-6 x-12))/(-6 (2 x^2+2 x-4))
Factor 2 out of 2 x^2+2 x-4:
((15 x^2-9 x-6) (6 x^2-6 x-12))/(-62 (x^2+x-2))
The factors of -2 that sum to are 2 and -1. So, x^2+x-2 = (x+2) (x-1):
((15 x^2-9 x-6) (6 x^2-6 x-12))/(-6×2 (x+2) (x-1))
Factor 6 out of 6 x^2-6 x-12:
(6 (x^2-x-2) (15 x^2-9 x-6))/(-6×2 (x+2) (x-1))
The factors of -2 that sum to -1 are 1 and -2. So, x^2-x-2 = (x+1) (x-2):
(6 (x+1) (x-2) (15 x^2-9 x-6))/(-6×2 (x+2) (x-1))
Factor 3 out of 15 x^2-9 x-6:
(6×3 (5 x^2-3 x-2) (x+1) (x-2))/(-6×2 (x+2) (x-1))
Factor the quadratic 5 x^2-3 x-2. The coefficient of x^2 is 5 and the constant term is -2. The product of 5 and -2 is -10. The factors of -10 which sum to -3 are 2 and -5. So 5 x^2-3 x-2 = 5 x^2-5 x+2 x-2 = x (5 x+2)-(5 x+2):
(3×6 x (5 x+2)-(5 x+2) (x+1) (x-2))/(-6×2 (x+2) (x-1))
Factor 5 x+2 from x (5 x+2)-(5 x+2):
(3×6 (5 x+2) (x-1) (x+1) (x-2))/(-6×2 (x+2) (x-1))
(3 (5 x+2) (x-1)×6 (x+1) (x-2))/(-6×2 (x+2) (x-1)) = (x-1)/(x-1)×(3 (5 x+2)×6 (x+1) (x-2))/(-6×2 (x+2)) = (3 (5 x+2)×6 (x+1) (x-2))/(-6×2 (x+2)):
(3×6 (5 x+2) (x+1) (x-2))/(-6×2 (x+2))
6/(-6) = 6/(6 (-1)) = 1/(-1):
(3 (5 x+2) (x+1) (x-2))/(-1×2 (x+2))
Multiply numerator and denominator of (3 (5 x+2) (x+1) (x-2))/(-2 (x+2)) by -1:
Answer: |-(3 (5x+2) (x+1) (x-2))/(2 (x+2))
