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Difficulty: HardSimplifying Expressions and Combining Like Terms

For each algebraic expression on the left, match it to its completely simplified equivalent expression on the right by distributing terms and combining like terms.

  • 2x(x2−3xy)−3y(x2−y2)−(2x3−6x2y)2x(x^2 - 3xy) - 3y(x^2 - y^2) - (2x^3 - 6x^2y)−3x2y+3y3-3x^2y + 3y^3
  • (2x−y)3−8x(x2−3xy)−y3(2x - y)^3 - 8x(x^2 - 3xy) - y^312x2y+6xy2−2y312x^2y + 6xy^2 - 2y^3
  • x(2x−3y)2−y(x−2y)2−(4x3−13x2y)x(2x - 3y)^2 - y(x - 2y)^2 - (4x^3 - 13x^2y)13xy2−4y313xy^2 - 4y^3
  • 2x2(x−3y)−(x−y)3−y2(3x−y)2x^2(x - 3y) - (x - y)^3 - y^2(3x - y)x3−3x2y−6xy2+2y3x^3 - 3x^2y - 6xy^2 + 2y^3

Answer

The expressions match as follows: the first simplifies to −3x2y+3y3-3x^2y + 3y^3; the second simplifies to 12x2y+6xy2−2y312x^2y + 6xy^2 - 2y^3; the third simplifies to 13xy2−4y313xy^2 - 4y^3; and the fourth simplifies to x3−3x2y−6xy2+2y3x^3 - 3x^2y - 6xy^2 + 2y^3.
Each expression is expanded fully by distributing multiplication and powers, then simplified by combining terms that have the exact same variable bases and exponents.

Step-by-Step Solution

1
Simplify the first expression by distributing coefficients and combining like terms.
−3x2y+3y3-3x^2y + 3y^3
Expanding the expression gives 2x3−6x2y−3x2y+3y3−2x3+6x2y2x^3 - 6x^2y - 3x^2y + 3y^3 - 2x^3 + 6x^2y. Grouping the like terms: (2−2)x3+(−6−3+6)x2y+3y3(2 - 2)x^3 + (-6 - 3 + 6)x^2y + 3y^3, which simplifies to −3x2y+3y3-3x^2y + 3y^3.
2
Simplify the second expression using the binomial cube formula (a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.
12x2y+6xy2−2y312x^2y + 6xy^2 - 2y^3
Expanding (2x−y)3(2x - y)^3 yields 8x3−12x2y+6xy2−y38x^3 - 12x^2y + 6xy^2 - y^3. Subtracting the remaining terms gives 8x3−12x2y+6xy2−y3−8x3+24x2y−y38x^3 - 12x^2y + 6xy^2 - y^3 - 8x^3 + 24x^2y - y^3. Combining like terms yields 12x2y+6xy2−2y312x^2y + 6xy^2 - 2y^3.
3
Simplify the third expression by squaring the binomials and distributing.
13xy2−4y313xy^2 - 4y^3
First expand the squares: (2x−3y)2=4x2−12xy+9y2(2x - 3y)^2 = 4x^2 - 12xy + 9y^2 and (x−2y)2=x2−4xy+4y2(x - 2y)^2 = x^2 - 4xy + 4y^2. Distributing the variables yields 4x3−12x2y+9xy2−x2y+4xy2−4y3−4x3+13x2y4x^3 - 12x^2y + 9xy^2 - x^2y + 4xy^2 - 4y^3 - 4x^3 + 13x^2y. Combining like terms results in 13xy2−4y313xy^2 - 4y^3.
4
Simplify the fourth expression by expanding the cubic term and distributing signs.
x3−3x2y−6xy2+2y3x^3 - 3x^2y - 6xy^2 + 2y^3
Expand (x−y)3=x3−3x2y+3xy2−y3(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3. Distribute all negative signs to get 2x3−6x2y−x3+3x2y−3xy2+y3−3xy2+y32x^3 - 6x^2y - x^3 + 3x^2y - 3xy^2 + y^3 - 3xy^2 + y^3. Combining like terms results in x3−3x2y−6xy2+2y3x^3 - 3x^2y - 6xy^2 + 2y^3.

Key Concept

Simplifying Expressions and Combining Like Terms
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Simplifying Expressions and Combining Like Terms

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