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Difficulty: Very hardSimplifying and Factoring Algebraic Expressions

Given non-zero real numbers xx and yy where ∣x∣≠∣y∣|x| \neq |y|, simplify the complex rational expression:

x3+y3x2−y2−x2y−xy2(x−y)2x4−y4x3+x2y+xy2+y3\frac{\frac{x^3 + y^3}{x^2 - y^2} - \frac{x^2y - xy^2}{(x - y)^2}}{\frac{x^4 - y^4}{x^3 + x^2y + xy^2 + y^3}}

Which of the following represents the completely simplified expression?

  1. 1Answer
  2. B
    x - y
  3. C
    x + y
  4. D
    x−yx+y\frac{x - y}{x + y}
  5. E
    x2+y2x−y\frac{x^2 + y^2}{x - y}

Answer

1
Both the entire complex numerator and the entire complex denominator independently simplify to x−yx - y. Consequently, dividing the numerator x−yx - y by the denominator x−yx - y gives 11.

Step-by-Step Solution

1
Simplify the first term of the main numerator
\frac{x^3 + y^3}{x^2 - y^2} = \frac{(x + y)(x^2 - xy + y^2)}{(x - y)(x + y)} = \frac{x^2 - xy + y^2}{x - y}
Factor the sum of cubes in the numerator and the difference of squares in the denominator, then cancel the common factor (x+y)(x + y).
2
Simplify the second term of the main numerator
\frac{x^2y - xy^2}{(x - y)^2} = \frac{xy(x - y)}{(x - y)^2} = \frac{xy}{x - y}
Factor out the greatest common factor xyxy from the numerator and cancel one factor of (x−y)(x - y).
3
Subtract the simplified terms in the main numerator
\frac{x^2 - xy + y^2}{x - y} - \frac{xy}{x - y} = \frac{x^2 - 2xy + y^2}{x - y} = \frac{(x - y)^2}{x - y} = x - y
Combine the numerators over the common denominator (x−y)(x - y), factor the perfect square trinomial x2−2xy+y2=(x−y)2x^2 - 2xy + y^2 = (x - y)^2, and simplify.
4
Simplify the main denominator
\frac{x^4 - y^4}{x^3 + x^2y + xy^2 + y^3} = \frac{(x - y)(x + y)(x^2 + y^2)}{x^2(x + y) + y^2(x + y)} = \frac{(x - y)(x + y)(x^2 + y^2)}{(x + y)(x^2 + y^2)} = x - y
Factor the numerator using difference of squares twice, factor the denominator by grouping, and cancel common factors (x+y)(x2+y2)(x + y)(x^2 + y^2).
5
Divide the main numerator by the main denominator
\frac{x - y}{x - y} = 1
Divide the simplified main numerator (x−yx - y) by the simplified main denominator (x−yx - y).

Key Concept

Multi-step algebraic expression simplification using special factoring identities (sum/difference of cubes, difference of squares, quadratic trinomials, and factoring by grouping).
Estimated Time:2m 0s
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Topics

Simplifying and Factoring Algebraic Expressions

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