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Difficulty: EasySimplifying Expressions and Combining Like Terms

For all real values of xx and yy, which of the following is equivalent to the expression x(x−y)2−x2(x−2y)x(x - y)^2 - x^2(x - 2y)?

  1. xy2xy^2Answer
  2. B
    2x2y+xy22x^2y + xy^2
  3. C
    −4x2y+xy2-4x^2y + xy^2
  4. D
    −x3+x2+xy2-x^3 + x^2 + xy^2
  5. E
    y2y^2

Answer

The simplified expression is xy2xy^2.
Expanding (x−y)2(x - y)^2 yields x2−2xy+y2x^2 - 2xy + y^2. Distributing xx to this expression results in x3−2x2y+xy2x^3 - 2x^2y + xy^2. Distributing −x2-x^2 to (x−2y)(x - 2y) yields −x3+2x2y-x^3 + 2x^2y. Combining these parts gives (x3−x3)+(−2x2y+2x2y)+xy2(x^3 - x^3) + (-2x^2y + 2x^2y) + xy^2, which simplifies completely to xy2xy^2.

Step-by-Step Solution

1
Expand the squared binomial (x−y)2(x - y)^2 using the algebraic identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
(x−y)2=x2−2xy+y2(x - y)^2 = x^2 - 2xy + y^2
Expanding the binomial is necessary before distributing the outer variable.
2
Distribute the term xx to each term in the expanded binomial, and distribute the term −x2-x^2 to each term inside the second parenthesis.
x(x2−2xy+y2)=x3−2x2y+xy2x(x^2 - 2xy + y^2) = x^3 - 2x^2y + xy^2 and −x2(x−2y)=−x3+2x2y-x^2(x - 2y) = -x^3 + 2x^2y
Distribution eliminates parentheses and prepares the expression for combining like terms.
3
Combine all like terms in the resulting expression: (x3−x3)+(−2x2y+2x2y)+xy2(x^3 - x^3) + (-2x^2y + 2x^2y) + xy^2.
xy2xy^2
Combining like terms simplifies the expression to its final equivalent form.

Key Concept

Simplifying algebraic expressions by expanding binomials, distributing variables, and combining like terms.
Estimated Time:1m 0s
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Topics

Simplifying Expressions and Combining Like Terms

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