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Difficulty: EasySimplifying Expressions and Combining Like Terms

For all real numbers xx and yy, which of the following is equivalent to the expression (x−2y)2−(x2−5xy)(x - 2y)^2 - (x^2 - 5xy)?

  1. A
    4y+xy4y + xy
  2. B
    4y2−9xy4y^2 - 9xy
  3. C
    4y2+5xy4y^2 + 5xy
  4. 4y2+xy4y^2 + xyAnswer
  5. E
    4y2−5xy4y^2 - 5xy

Answer

The expression is equivalent to 4y2+xy4y^2 + xy.
Expanding (x−2y)2(x - 2y)^2 yields x2−4xy+4y2x^2 - 4xy + 4y^2. Distributing the negative sign to −(x2−5xy)-(x^2 - 5xy) gives −x2+5xy-x^2 + 5xy. Combining the results yields (x2−x2)+(−4xy+5xy)+4y2=xy+4y2(x^2 - x^2) + (-4xy + 5xy) + 4y^2 = xy + 4y^2, which is equivalent to 4y2+xy4y^2 + xy.

Step-by-Step Solution

1
Expand the squared binomial (x−2y)2(x - 2y)^2.
x2−4xy+4y2x^2 - 4xy + 4y^2
Apply the binomial squaring formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 to expand the expression.
2
Distribute the negative sign across the second parenthetical expression −(x2−5xy)-(x^2 - 5xy).
−x2+5xy-x^2 + 5xy
Distributing the negative sign changes the signs of both terms inside the parentheses.
3
Combine like terms from the expanded parts.
xy+4y2xy + 4y^2
Group and add coefficients of the like terms: (x2−x2)+(−4xy+5xy)+4y2=0+xy+4y2(x^2 - x^2) + (-4xy + 5xy) + 4y^2 = 0 + xy + 4y^2.

Key Concept

Simplifying Expressions and Combining Like Terms
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Topics

Simplifying Expressions and Combining Like Terms

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