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Difficulty: MediumSimplifying Expressions and Combining Like Terms

When the expression 2x(3x−y)2−3y2(x−2y)−x2(18x−15y)2x(3x - y)^2 - 3y^2(x - 2y) - x^2(18x - 15y) is simplified by combining like terms, what is the coefficient of x2yx^2y?

  1. A
    -27
  2. B
    -12
  3. 3Answer
  4. D
    15
  5. E
    -15

Answer

The coefficient of x2yx^2y is 33.
The correct coefficient of 33 is obtained by expanding the expression step-by-step. First, the square of the binomial (3x−y)2(3x - y)^2 is 9x2−6xy+y29x^2 - 6xy + y^2. Distributing 2x2x gives 18x3−12x2y+2xy218x^3 - 12x^2y + 2xy^2. Next, distributing −x2-x^2 to (18x−15y)(18x - 15y) gives −18x3+15x2y-18x^3 + 15x^2y. Combining the x2yx^2y terms gives −12x2y+15x2y=3x2y-12x^2y + 15x^2y = 3x^2y, meaning the coefficient is 33.

Step-by-Step Solution

1
Expand the squared binomial term (3x−y)2(3x - y)^2 using the algebraic identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
(3x−y)2=9x2−6xy+y2(3x - y)^2 = 9x^2 - 6xy + y^2
This is necessary to remove the parentheses before distributing the outer variable.
2
Multiply the term 2x2x by each term inside the expanded binomial expression: 2x(9x2−6xy+y2)2x(9x^2 - 6xy + y^2).
18x3−12x2y+2xy218x^3 - 12x^2y + 2xy^2
Applying the distributive property expands the first part of the expression.
3
Distribute −x2-x^2 to the terms inside the parentheses (18x−15y)(18x - 15y), paying close attention to the signs.
−18x3+15x2y-18x^3 + 15x^2y
This expands the third part of the expression and correctly distributes the negative sign.
4
Identify and combine the like terms of the form x2yx^2y from the expanded parts of the expression.
−12x2y+15x2y=3x2y-12x^2y + 15x^2y = 3x^2y
To find the coefficient of x2yx^2y, we only need to sum the coefficients of the terms that contain exactly x2yx^2y.

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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