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Difficulty: HardSimplifying and Factoring Algebraic Expressions

For all non-zero real numbers xx and yy such that ∣x∣≠∣y∣|x| \neq |y| and x2+2xy+2y2≠0x^2 + 2xy + 2y^2 \neq 0, consider the algebraic expression:

Q(x,y)=x4+4y4x2+2xy+2y2+2x3y−2xy3x2−y2Q(x, y) = \frac{x^4 + 4y^4}{x^2 + 2xy + 2y^2} + \frac{2x^3 y - 2xy^3}{x^2 - y^2}

Which of the following expressions are equivalent to Q(x,y)Q(x, y)? Indicate all such expressions.

  1. (x−y)2+2xy+y2(x - y)^2 + 2xy + y^2Answer
  2. x3+2xy2x\frac{x^3 + 2xy^2}{x}Answer
  3. C
    \frac{(x^2 + 2y^2)^2 - 4x^2 y^2}{x^2 - 2xy + 2y^2}
  4. D
    x4−4y4x2+2y2\frac{x^4 - 4y^4}{x^2 + 2y^2}
  5. E
    \frac{x^4 + 2x^2 y^2 + 2y^4}{x^2 + y^2}

Answer

The expressions equivalent to Q(x,y)Q(x, y) are (x−y)2+2xy+y2(x - y)^2 + 2xy + y^2 and x3+2xy2x\frac{x^3 + 2xy^2}{x}.
First, simplify Q(x,y)Q(x, y) by factoring each term. Using Sophie Germain's identity on the numerator of the first term gives x4+4y4=(x2+2y2)2−(2xy)2=(x2+2xy+2y2)(x2−2xy+2y2)x^4 + 4y^4 = (x^2 + 2y^2)^2 - (2xy)^2 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2). Dividing this by (x2+2xy+2y2)(x^2 + 2xy + 2y^2) results in x2−2xy+2y2x^2 - 2xy + 2y^2. For the second term, factoring out 2xy2xy gives 2xy(x2−y2)x2−y2=2xy\frac{2xy(x^2 - y^2)}{x^2 - y^2} = 2xy. Adding the two simplified terms yields Q(x,y)=(x2−2xy+2y2)+2xy=x2+2y2Q(x, y) = (x^2 - 2xy + 2y^2) + 2xy = x^2 + 2y^2.

Evaluating the options for equivalence:
- The expression (x−y)2+2xy+y2(x - y)^2 + 2xy + y^2 expands to x2−2xy+y2+2xy+y2=x2+2y2x^2 - 2xy + y^2 + 2xy + y^2 = x^2 + 2y^2.
- The expression x3+2xy2x\frac{x^3 + 2xy^2}{x} factors as x(x2+2y2)x=x2+2y2\frac{x(x^2 + 2y^2)}{x} = x^2 + 2y^2.
Therefore, both of these expressions are equivalent to Q(x,y)Q(x, y).

Step-by-Step Solution

1
Simplify the first algebraic fraction using Sophie Germain's identity.
x4+4y4x2+2xy+2y2=(x2+2y2)2−(2xy)2x2+2xy+2y2=(x2+2xy+2y2)(x2−2xy+2y2)x2+2xy+2y2=x2−2xy+2y2\frac{x^4 + 4y^4}{x^2 + 2xy + 2y^2} = \frac{(x^2 + 2y^2)^2 - (2xy)^2}{x^2 + 2xy + 2y^2} = \frac{(x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2)}{x^2 + 2xy + 2y^2} = x^2 - 2xy + 2y^2
Completing the square on x4+4y4x^4 + 4y^4 allows it to be factored into the product of two quadratic expressions.
2
Simplify the second algebraic fraction by factoring out common factors.
2x3y−2xy3x2−y2=2xy(x2−y2)x2−y2=2xy\frac{2x^3 y - 2xy^3}{x^2 - y^2} = \frac{2xy(x^2 - y^2)}{x^2 - y^2} = 2xy
The term (x2−y2)(x^2 - y^2) cancels out since x≠±yx \neq \pm y.
3
Combine the simplified terms to find the closed-form expression for Q(x,y)Q(x, y).
Q(x,y)=(x2−2xy+2y2)+2xy=x2+2y2Q(x, y) = (x^2 - 2xy + 2y^2) + 2xy = x^2 + 2y^2
The −2xy-2xy and +2xy+2xy terms sum to zero.
4
Test each option for equivalence to x2+2y2x^2 + 2y^2.
The option (x−y)2+2xy+y2(x - y)^2 + 2xy + y^2 expands to x2−2xy+y2+2xy+y2=x2+2y2x^2 - 2xy + y^2 + 2xy + y^2 = x^2 + 2y^2. The option x3+2xy2x\frac{x^3 + 2xy^2}{x} simplifies directly to x2+2y2x^2 + 2y^2.
Both expressions reduce identically to x2+2y2x^2 + 2y^2 for all non-zero xx and yy.

Key Concept

Simplifying complex algebraic expressions using polynomial factoring identities (Sophie Germain identity and difference of squares) and algebraic reduction.
Estimated Time:2m 0s
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Topics

Simplifying and Factoring Algebraic Expressions

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