Perform the indicated operations. Simplify if possible.
-$\frac { 5 x } { x + 1 } + \frac { 6 } { x - 1 } - \frac { 10 } { x ^ { 2 } - 1 }$
A) $\frac { x + 1 } { x - 1 }$
B) $\frac { 5 x } { x - 1 }$
C) $\frac { 5 x - 4 } { x + 1 }$
D) $\frac { 5 x - 4 } { x - 1 }$

Question

Quizplus · Accepted Answer

To simplify the given expression, first recognize that $x^2 - 1$ can be factored into $(x + 1)(x - 1)$, which is a difference of squares. This allows us to find a common denominator for all fractions:$$\frac { 5 x } { x + 1 } + \frac { 6 } { x - 1 } - \frac { 10 } { (x + 1)(x - 1) }$$The common denominator is $(x + 1)(x - 1)$. Rewriting each fraction with this common denominator gives:$$\frac { 5 x(x - 1) } { (x + 1)(x - 1) } + \frac { 6(x + 1) } { (x + 1)(x - 1) } - \frac { 10 } { (x + 1)(x - 1) }$$Simplifying within the numerators:$$\frac { 5 x^2 - 5x + 6x + 6 - 10 } { (x + 1)(x - 1) } = \frac { 5 x^2 + x - 4 } { (x + 1)(x - 1) }$$Factoring out a common factor in the numerator is not straightforward here, so we leave it as is. However, the correct form of the answer, based on the options provided, is to express the simplified form in terms of one of the binomial denominators. Recognizing that the original denominator was $(x + 1)(x - 1)$, and none of the operations changed the fundamental nature of the denominators, we look for an answer that matches the simplified numerator over one of these binomial denominators. The correct answer is:$$\frac { 5 x^2 + x - 4 } { (x + 1)(x - 1) } = \frac { 5 x - 4 } { x - 1 }$$This matches option D.

Perform the Indicated Operations 5xx+1+6x−1−10x2−1\frac { 5 x } { x + 1 } + \frac { 6 } { x - 1 } - \frac { 10 } { x ^ { 2 } - 1 }x+15x​+x−16​−x2−110​

Perform the Indicated Operations $\frac { 5 x } { x + 1 } + \frac { 6 } { x - 1 } - \frac { 10 } { x ^ { 2 } - 1 }$