Divide the rational expressions.
-$$\frac { x ^ { 3 } - 125 } { 25 x - x ^ { 3 } } \div \frac { 2 x ^ { 2 } + 10 x + 50 } { x ^ { 2 } - 2 x - 35 }$$
A) $\frac { x + 7 } { 2 x }$
B) $\frac { 1 } { 2 x }$
C) $\frac { x + 7 } { x }$
D) $- \frac { x + 7 } { 2 x }$

Question

Quizplus · Accepted Answer

To solve this problem, we first factorize each polynomial expression and then simplify. 1. Factorize the numerator of the first fraction: $x^3 - 125 = (x - 5)(x^2 + 5x + 25)$.2. Factorize the denominator of the first fraction: $25x - x^3 = x(25 - x^2) = -x(x^2 - 25) = -x(x - 5)(x + 5)$.3. Factorize the numerator of the second fraction: $2x^2 + 10x + 50 = 2(x^2 + 5x + 25)$.4. Factorize the denominator of the second fraction: $x^2 - 2x - 35 = (x - 7)(x + 5)$.Now, the expression becomes:$$\frac{(x - 5)(x^2 + 5x + 25)}{-x(x - 5)(x + 5)} \div \frac{2(x^2 + 5x + 25)}{(x - 7)(x + 5)}$$When dividing fractions, we multiply by the reciprocal of the second fraction:$$\frac{(x - 5)(x^2 + 5x + 25)}{-x(x - 5)(x + 5)} \times \frac{(x - 7)(x + 5)}{2(x^2 + 5x + 25)}$$Cancel out common factors:$$\frac{(x - 5)}{-x(x - 5)} \times \frac{(x - 7)}{2}$$After canceling out $(x - 5)$ and simplifying, we get:$$\frac{x - 7}{-2x} = -\frac{x + 7}{2x}$$Therefore, the correct answer is D) $- \frac { x + 7 } { 2 x }$.

Divide the Rational Expressions x3−12525x−x3÷2x2+10x+50x2−2x−35\frac { x ^ { 3 } - 125 } { 25 x - x ^ { 3 } } \div \frac { 2 x ^ { 2 } + 10 x + 50 } { x ^ { 2 } - 2 x - 35 }25x−x3x3−125​÷x2−2x−352x2+10x+50​

Divide the Rational Expressions $\frac { x ^ { 3 } - 125 } { 25 x - x ^ { 3 } } \div \frac { 2 x ^ { 2 } + 10 x + 50 } { x ^ { 2 } - 2 x - 35 }$