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Difficulty: Very hardSimplifying Expressions and Combining Like Terms

For all real numbers xx and yy, when the expression −2x(x−3y)2−y2(4x−y)−3x2(2x−y)-2x(x - 3y)^2 - y^2(4x - y) - 3x^2(2x - y) is completely simplified, what is the coefficient of the x2yx^2y term?

  1. A
    -9
  2. B
    3
  3. C
    9
  4. D
    12
  5. 15Answer

Answer

15
Completely simplifying the given expression yields −8x3+15x2y−22xy2+y3-8x^3 + 15x^2y - 22xy^2 + y^3. The coefficient of the x2yx^2y term is 1515, which is obtained by combining the term 12x2y12x^2y (from the distribution of −2x-2x over the middle term of the squared binomial) and the term 3x2y3x^2y (from the distribution of −3x2-3x^2 over −y-y).

Step-by-Step Solution

1
Expand the binomial squared expression (x−3y)2(x - 3y)^2.
(x−3y)2=x2−6xy+9y2(x - 3y)^2 = x^2 - 6xy + 9y^2
Following the order of operations, we must square the binomial before distributing the outer term.
2
Distribute −2x-2x to the trinomial result from the previous step.
−2x(x2−6xy+9y2)=−2x3+12x2y−18xy2-2x(x^2 - 6xy + 9y^2) = -2x^3 + 12x^2y - 18xy^2
Using the distributive property, we multiply coefficients and add exponents of like bases (noting that −2x×−6xy=12x2y-2x \times -6xy = 12x^2y).
3
Distribute −y2-y^2 to the binomial (4x−y)(4x - y).
−y2(4x−y)=−4xy2+y3-y^2(4x - y) = -4xy^2 + y^3
We multiply each term inside the parentheses by −y2-y^2, keeping track of the signs.
4
Distribute −3x2-3x^2 to the binomial (2x−y)(2x - y).
−3x2(2x−y)=−6x3+3x2y-3x^2(2x - y) = -6x^3 + 3x^2y
We distribute the −3x2-3x^2 factor, ensuring that multiplying two negative values results in a positive term (−3x2×−y=3x2y-3x^2 \times -y = 3x^2y).
5
Combine the coefficients of all like terms containing x2yx^2y.
12x2y+3x2y=15x2y12x^2y + 3x^2y = 15x^2y
We add the coefficients of the terms that share the exact variable part x2yx^2y to find the final coefficient.

Key Concept

Simplifying algebraic expressions by distributing terms, applying exponent rules, and combining like terms.
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Topics

Simplifying Expressions and Combining Like Terms

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