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Difficulty: Very hardSimplifying Expressions and Combining Like Terms

Simplify each of the algebraic expressions on the left by distributing and combining like terms, then match it with its equivalent simplified expression on the right.

  • −(x−2y)3−x(x+2y)(x−2y)+4y2(2x−y)-(x - 2y)^3 - x(x + 2y)(x - 2y) + 4y^2(2x - y)−2x3+6x2y+4y3-2x^3 + 6x^2y + 4y^3
  • 2x(x−y)2−(x−2y)(x2+2xy+4y2)−2xy(x−2y)2x(x - y)^2 - (x - 2y)(x^2 + 2xy + 4y^2) - 2xy(x - 2y)x3−6x2y+6xy2+8y3x^3 - 6x^2y + 6xy^2 + 8y^3
  • (x+y)3−(x−y)3−2y(3x2+y2)(x + y)^3 - (x - y)^3 - 2y(3x^2 + y^2)00
  • (x2−y)2−(x2+y)(x2−y)−y(y−2x2)(x^2 - y)^2 - (x^2 + y)(x^2 - y) - y(y - 2x^2)y2y^2

Answer

The expressions match as follows: the first expression matches −2x3+6x2y+4y3-2x^3 + 6x^2y + 4y^3; the second expression matches x3−6x2y+6xy2+8y3x^3 - 6x^2y + 6xy^2 + 8y^3; the third expression matches 00; and the fourth expression matches y2y^2.
Each expression on the left reduces to its matching counterpart on the right by carefully expanding terms (including cubes, squares, difference of squares, and difference of cubes) and combining like terms while correctly distributing negative signs.

Step-by-Step Solution

1
Simplify the first expression by expanding each term individually.
E1=−x3+6x2y−12xy2+8y3−x3+4xy2+8xy2−4y3E_1 = -x^3 + 6x^2y - 12xy^2 + 8y^3 - x^3 + 4xy^2 + 8xy^2 - 4y^3
Expanding the binomial cube, the difference of squares product, and distributing the monomial allows us to identify like terms.
2
Combine like terms in the first expression.
E1=−2x3+6x2y+4y3E_1 = -2x^3 + 6x^2y + 4y^3
Combining the x3x^3, x2yx^2y, xy2xy^2, and y3y^3 terms simplifies the expression. The xy2xy^2 terms sum to zero.
3
Simplify the second expression by expanding each term individually.
E2=2x3−4x2y+2xy2−x3+8y3−2x2y+4xy2E_2 = 2x^3 - 4x^2y + 2xy^2 - x^3 + 8y^3 - 2x^2y + 4xy^2
Using algebraic expansion rules (including the difference of cubes product) exposes all individual terms.
4
Combine like terms in the second expression.
E2=x3−6x2y+6xy2+8y3E_2 = x^3 - 6x^2y + 6xy^2 + 8y^3
Adding coefficients of like variable terms gives the simplified form.
5
Simplify the third expression by expanding the cubic terms.
E3=(x3+3x2y+3xy2+y3)−(x3−3x2y+3xy2−y3)−(6x2y+2y3)=6x2y+2y3−6x2y−2y3E_3 = (x^3 + 3x^2y + 3xy^2 + y^3) - (x^3 - 3x^2y + 3xy^2 - y^3) - (6x^2y + 2y^3) = 6x^2y + 2y^3 - 6x^2y - 2y^3
Expanding the binomial cubes and distributing the negative signs shows that all terms cancel.
6
Combine like terms in the third expression.
E3=0E_3 = 0
All terms cancel out, leaving a final value of zero.
7
Simplify the fourth expression by expanding.
E4=x4−2x2y+y2−x4+y2−y2+2x2yE_4 = x^4 - 2x^2y + y^2 - x^4 + y^2 - y^2 + 2x^2y
Squaring the binomial, using the difference of squares, and distributing the negative variable simplifies the individual components.
8
Combine like terms in the fourth expression.
E4=y2E_4 = y^2
The x4x^4 and x2yx^2y terms cancel, and the y2y^2 terms simplify to y2y^2.

Key Concept

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

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