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Difficulty: HardSimplifying Algebraic Expressions

Match each complex algebraic expression with its completely simplified polynomial form.

  • 3x(x−2y)−[2x2−x(4y−3x)]3x(x - 2y) - [2x^2 - x(4y - 3x)]−2x2−2xy-2x^2 - 2xy
  • (2x−y)2−4x(x−y)(2x - y)^2 - 4x(x - y)y2y^2
  • −(x−3y)(x+3y)+x(x−2y)-(x - 3y)(x + 3y) + x(x - 2y)−2xy+9y2-2xy + 9y^2
  • 4xy−2[x(y−x)+y(x+2y)]4xy - 2[x(y - x) + y(x + 2y)]2x2−4y22x^2 - 4y^2

Answer

The first expression maps to −2x2−2xy-2x^2 - 2xy, the second expression maps to y2y^2, the third expression maps to −2xy+9y2-2xy + 9y^2, and the fourth expression maps to 2x2−4y22x^2 - 4y^2.
Each expression represents a distinct structural archetype for polynomial simplification. By systematically applying the order of operations—expanding binomials, distributing negative coefficients, and resolving nested grouping symbols from the inside out—all like terms can be properly combined to reach the correct canonical form.

Step-by-Step Solution

1
Simplify the first expression by resolving innermost parentheses, then brackets, then combining terms.
3x2−6xy−[2x2−4xy+3x2]3x^2 - 6xy - [2x^2 - 4xy + 3x^2] simplifies to 3x2−6xy−5x2+4xy3x^2 - 6xy - 5x^2 + 4xy, which yields −2x2−2xy-2x^2 - 2xy.
Following the order of operations from the innermost grouping symbols prevents sign errors when distributing negative coefficients.
2
Simplify the second expression by expanding the binomial square and distributing.
(4x2−4xy+y2)−(4x2−4xy)(4x^2 - 4xy + y^2) - (4x^2 - 4xy) simplifies directly to y2y^2.
Proper binomial expansion (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2 avoids the common error of dropping the middle term, allowing opposite terms to cancel.
3
Simplify the third expression using the difference of squares formula, then distribute the negative.
−(x2−9y2)+x2−2xy-(x^2 - 9y^2) + x^2 - 2xy simplifies to −x2+9y2+x2−2xy-x^2 + 9y^2 + x^2 - 2xy, which reduces to −2xy+9y2-2xy + 9y^2.
Recognizing the difference of squares simplifies multiplication, and properly distributing the outer negative sign ensures the squared variable term has the correct positive sign.
4
Simplify the fourth expression by distributing within the brackets before multiplying by the outer coefficient.
4xy−2[−x2+2xy+2y2]4xy - 2[-x^2 + 2xy + 2y^2] simplifies to 4xy+2x2−4xy−4y2=2x2−4y24xy + 2x^2 - 4xy - 4y^2 = 2x^2 - 4y^2.
Combining like terms inside the bracket before multiplying by the outer constant significantly reduces the chance of sign and variable combination errors.

Key Concept

Simplifying Complex Multivariable Algebraic Expressions
Estimated Time:3m 0s
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Topics

Simplifying Algebraic Expressions

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