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Difficulty: MediumSimplifying Expressions and Combining Like Terms

Match each unsimplified algebraic expression on the left with its equivalent simplified form on the right.

  • 3x(x−2y)−2x(x−3y)3x(x - 2y) - 2x(x - 3y)x2x^2
  • 2(x2−xy)−3(xy−y2)2(x^2 - xy) - 3(xy - y^2)2x2−5xy+3y22x^2 - 5xy + 3y^2
  • x2(x−2y)−x(x2−xy)x^2(x - 2y) - x(x^2 - xy)−x2y-x^2y
  • (x+y)2−(x−y)2(x + y)^2 - (x - y)^24xy4xy

Answer

The correct pairings match 3x(x−2y)−2x(x−3y)3x(x - 2y) - 2x(x - 3y) to x2x^2; 2(x2−xy)−3(xy−y2)2(x^2 - xy) - 3(xy - y^2) to 2x2−5xy+3y22x^2 - 5xy + 3y^2; x2(x−2y)−x(x2−xy)x^2(x - 2y) - x(x^2 - xy) to −x2y-x^2y; and (x+y)2−(x−y)2(x + y)^2 - (x - y)^2 to 4xy4xy.
Each unsimplified expression is correctly matched to its simplified equivalent by expanding parenthetical terms (taking care to distribute negative signs) and combining like terms.

Step-by-Step Solution

1
Distribute and combine like terms for the first expression 3x(x−2y)−2x(x−3y)3x(x - 2y) - 2x(x - 3y).
The expression simplifies to x2x^2.
First distribute the coefficients to get 3x2−6xy−2x2+6xy3x^2 - 6xy - 2x^2 + 6xy. Then combine 3x2−2x2=x23x^2 - 2x^2 = x^2 and −6xy+6xy=0-6xy + 6xy = 0.
2
Distribute and combine like terms for the second expression 2(x2−xy)−3(xy−y2)2(x^2 - xy) - 3(xy - y^2).
The expression simplifies to 2x2−5xy+3y22x^2 - 5xy + 3y^2.
Distribute the coefficients to get 2x2−2xy−3xy+3y22x^2 - 2xy - 3xy + 3y^2. Note that distributing the negative sign of −3-3 to −y2-y^2 results in +3y2+3y^2. Then combine −2xy−3xy=−5xy-2xy - 3xy = -5xy.
3
Distribute and combine like terms for the third expression x2(x−2y)−x(x2−xy)x^2(x - 2y) - x(x^2 - xy).
The expression simplifies to −x2y-x^2y.
Distribute to get x3−2x2y−x3+x2yx^3 - 2x^2y - x^3 + x^2y. Note that distributing −x-x to −xy-xy gives +x2y+x^2y. The x3−x3x^3 - x^3 terms cancel, leaving −2x2y+x2y=−x2y-2x^2y + x^2y = -x^2y.
4
Expand and simplify the fourth expression (x+y)2−(x−y)2(x + y)^2 - (x - y)^2.
The expression simplifies to 4xy4xy.
Expand both squared binomials: (x2+2xy+y2)−(x2−2xy+y2)(x^2 + 2xy + y^2) - (x^2 - 2xy + y^2). Distribute the negative sign to get x2+2xy+y2−x2+2xy−y2x^2 + 2xy + y^2 - x^2 + 2xy - y^2. Combine terms to cancel x2x^2 and y2y^2, leaving 4xy4xy.

Key Concept

Simplifying Expressions and Combining Like Terms
Estimated Time:1m 30s
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Topics

Simplifying Expressions and Combining Like Terms

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