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Question

Difficulty: MediumRational and Radical Expressions and Equations

For all real numbers xx such that x≠1x \neq 1 and x≠−1x \neq -1, which of the following is equivalent to the expression (1x−1+1x+1)2\left(\frac{1}{x-1} + \frac{1}{x+1}\right)^2?

  1. A
    1x2\frac{1}{x^2}
  2. B
    2x2(x2−1)2\frac{2x^2}{(x^2-1)^2}
  3. C
    2x2+2(x2−1)2\frac{2x^2+2}{(x^2-1)^2}
  4. D
    4x2x4−1\frac{4x^2}{x^4-1}
  5. 4x2(x2−1)2\frac{4x^2}{(x^2-1)^2}Answer

Answer

The expression is equivalent to the fraction with numerator four times x squared and denominator the square of the difference x squared minus one.
The correct answer is found by first rewriting the terms inside the parentheses with the common denominator (x2−1)(x^2-1), which simplifies the sum to 2xx2−1\frac{2x}{x^2-1}. Squaring the numerator (2x)2(2x)^2 yields 4x24x^2, and squaring the denominator yields (x2−1)2(x^2-1)^2.

Step-by-Step Solution

1
Find a common denominator for the two fractions inside the parentheses.
The common denominator is (x−1)(x+1)=x2−1(x-1)(x+1) = x^2-1. The expression inside the parentheses becomes x+1x2−1+x−1x2−1\frac{x+1}{x^2-1} + \frac{x-1}{x^2-1}.
To add rational expressions, they must have a common denominator.
2
Combine the numerators over the common denominator.
The sum is (x+1)+(x−1)x2−1=2xx2−1\frac{(x+1) + (x-1)}{x^2-1} = \frac{2x}{x^2-1}.
Combine the like terms in the numerator to simplify the expression before squaring.
3
Square the simplified rational expression by squaring the numerator and the denominator separately.
(2xx2−1)2=(2x)2(x2−1)2=4x2(x2−1)2.\left(\frac{2x}{x^2-1}\right)^2 = \frac{(2x)^2}{(x^2-1)^2} = \frac{4x^2}{(x^2-1)^2}.
Apply the power of a quotient property to obtain the final simplified expression.

Key Concept

Simplifying rational expressions and applying the power of a quotient rule.
Estimated Time:1m 30s
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Topics

Rational and Radical Expressions and Equations

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