Final answer:
To simplify the given expression, we factor where possible and then cancel out common factors. After this process, the expression simplifies to (18)/(x²-7x).
Step-by-step explanation:
To fully simplify the expression (6x³+12x²-210x)/(x²-49) × (6x)/(2x³-50x) ÷ (x+7)/(x²-49), we should first factor the polynomials where possible and cancel out common factors. Notice that x² - 49 is a difference of squares which can be factored into (x+7)(x-7). Additionally, the numerator 6x³+12x²-210x can be factored out by 6x, resulting in 6x(x²+2x-35), and x²+2x-35 can further be factored into (x+7)(x-5). Expression simplification involves canceling out common factors in the numerator and the denominator.
The expression simplifies as follows:
- Factor where possible
- Cancel out common factors
- Simplify the remaining terms
After factoring, the expression becomes:
(6x(x+7)(x-5)/(x+7)(x-7)) × (6x)/(2x(x²-25)) ÷ (x+7)/(x+7)(x-7)
We can now simplify:
(6x(x-5)/(x-7)) × (3)/(x(x-5)) ÷ 1/(x-7)
And finally:
(18)/(x²-7x)
This is a simplified expression.