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Difficulty: MediumSimplifying Expressions and Combining Like Terms

When the expression 2x2(3x−y)−3y(x2−2y)−(x3−3xy2)2x^2(3x - y) - 3y(x^2 - 2y) - (x^3 - 3xy^2) is fully simplified by combining like terms, which of the following represents the resulting expression?

  1. 5x3−5x2y+3xy2+6y25x^3 - 5x^2y + 3xy^2 + 6y^2Answer
  2. B
    5x3−5x2y−3xy2−6y25x^3 - 5x^2y - 3xy^2 - 6y^2
  3. C
    −x3+6x2−5x2y+3xy2+6y2-x^3 + 6x^2 - 5x^2y + 3xy^2 + 6y^2
  4. D
    5x3−5x2y+3xy+6y5x^3 - 5x^2y + 3xy + 6y
  5. E
    5x3−x2y+3xy2+6y25x^3 - x^2y + 3xy^2 + 6y^2

Answer

5x3−5x2y+3xy2+6y25x^3 - 5x^2y + 3xy^2 + 6y^2
The correct expression is obtained by systematically distributing the coefficients outside the parentheses and then grouping and combining the coefficients of the like terms: (6x3−x3)+(−2x2y−3x2y)+3xy2+6y2=5x3−5x2y+3xy2+6y2(6x^3 - x^3) + (-2x^2y - 3x^2y) + 3xy^2 + 6y^2 = 5x^3 - 5x^2y + 3xy^2 + 6y^2.

Step-by-Step Solution

1
Distribute 2x22x^2 to each term in the first parenthetical expression: (3x−y)(3x - y)
6x3−2x2y6x^3 - 2x^2y
Applying the distributive property of multiplication over subtraction.
2
Distribute −3y-3y to each term in the second parenthetical expression: (x2−2y)(x^2 - 2y)
−3x2y+6y2-3x^2y + 6y^2
Applying the distributive property and multiplying negative coefficients (−3×−2=6-3 \times -2 = 6).
3
Distribute the negative sign to each term in the third parenthetical expression: (x3−3xy2)(x^3 - 3xy^2)
−x3+3xy2-x^3 + 3xy^2
Distributing −1-1 across the parentheses to remove them.
4
Combine the expanded expressions and group the like terms
(6x3−x3)+(−2x2y−3x2y)+3xy2+6y2(6x^3 - x^3) + (-2x^2y - 3x^2y) + 3xy^2 + 6y^2
Grouping together terms that have the same variables raised to the same powers.
5
Simplify by performing the operations on the coefficients of the like terms
5x3−5x2y+3xy2+6y25x^3 - 5x^2y + 3xy^2 + 6y^2
Combining coefficients: 6−1=56 - 1 = 5 for the x3x^3 terms, and −2−3=−5-2 - 3 = -5 for the x2yx^2y terms.

Key Concept

Simplifying algebraic expressions by distributing coefficients and combining like terms.

Alternative Method

To check your work, substitute simple values for xx and yy, such as x=1x = 1 and y=1y = 1, into the original expression and the simplified expression. Evaluating the original expression: 2(1)2(3(1)−1)−3(1)(12−2(1))−(13−3(1)(1)2)=2(2)−3(−1)−(1−3)=4+3−(−2)=92(1)^2(3(1) - 1) - 3(1)(1^2 - 2(1)) - (1^3 - 3(1)(1)^2) = 2(2) - 3(-1) - (1 - 3) = 4 + 3 - (-2) = 9. Evaluating the correct simplified expression: 5(1)3−5(1)2(1)+3(1)(1)2+6(1)2=5−5+3+6=95(1)^3 - 5(1)^2(1) + 3(1)(1)^2 + 6(1)^2 = 5 - 5 + 3 + 6 = 9. Since both evaluations yield 9, this confirms the simplification.
Estimated Time:1m 0s
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Simplifying Expressions and Combining Like Terms

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