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Difficulty: HardSimplifying Expressions and Combining Like Terms

A business analyst models the daily cost, CC, and daily revenue, RR, of a manufacturing process using the following expressions:

C=2x2(x−3y)−y(x2−2y2)C = 2x^2(x - 3y) - y(x^2 - 2y^2)
R=5x3−x(y−2x)2+3y3R = 5x^3 - x(y - 2x)^2 + 3y^3

where xx represents the number of units of product X sold, and yy represents the number of units of product Y sold. The daily profit, PP, is defined as P=R−CP = R - C. When the profit expression is fully simplified by combining like terms, which of the following expressions represents the daily profit PP?

  1. A
    $-x^3 + 7x^2y - xy^2 + y^3
  2. B
    $-x^3 + 11x^2y - xy^2 + 5y^3
  3. $-x^3 + 11x^2y - xy^2 + y^3Answer
  4. D
    $-x^3 + 3x^2y - xy^2 + y^3
  5. E
    $3x^3 - 4x^2 + 11x^2y - xy^2 + y^3

Answer

−x3+11x2y−xy2+y3-x^3 + 11x^2y - xy^2 + y^3
The expression −x3+11x2y−xy2+y3-x^3 + 11x^2y - xy^2 + y^3 is correct. Calculating profit requires subtracting the fully simplified cost from the fully simplified revenue. Simplifying CC yields 2x3−7x2y+2y32x^3 - 7x^2y + 2y^3. Expanding and simplifying RR yields x3+4x2y−xy2+3y3x^3 + 4x^2y - xy^2 + 3y^3. Subtracting these two expressions and correctly distributing the negative sign gives −x3+11x2y−xy2+y3-x^3 + 11x^2y - xy^2 + y^3.

Step-by-Step Solution

1
Expand and simplify the cost expression C=2x2(x−3y)−y(x2−2y2)C = 2x^2(x - 3y) - y(x^2 - 2y^2).
C=2x3−6x2y−x2y+2y3=2x3−7x2y+2y3C = 2x^3 - 6x^2y - x^2y + 2y^3 = 2x^3 - 7x^2y + 2y^3
Distribute the terms outside the parentheses and combine the like terms −6x2y-6x^2y and −x2y-x^2y.
2
Expand and simplify the revenue expression R=5x3−x(y−2x)2+3y3R = 5x^3 - x(y - 2x)^2 + 3y^3.
R=5x3−x(y2−4xy+4x2)+3y3=5x3−xy2+4x2y−4x3+3y3=x3+4x2y−xy2+3y3R = 5x^3 - x(y^2 - 4xy + 4x^2) + 3y^3 = 5x^3 - xy^2 + 4x^2y - 4x^3 + 3y^3 = x^3 + 4x^2y - xy^2 + 3y^3
Square the binomial (y−2x)2=y2−4xy+4x2(y - 2x)^2 = y^2 - 4xy + 4x^2, distribute the −x-x, and combine the like terms 5x35x^3 and −4x3-4x^3.
3
Subtract the cost expression from the revenue expression to find P=R−CP = R - C.
P=(x3+4x2y−xy2+3y3)−(2x3−7x2y+2y3)=x3+4x2y−xy2+3y3−2x3+7x2y−2y3=−x3+11x2y−xy2+y3P = (x^3 + 4x^2y - xy^2 + 3y^3) - (2x^3 - 7x^2y + 2y^3) = x^3 + 4x^2y - xy^2 + 3y^3 - 2x^3 + 7x^2y - 2y^3 = -x^3 + 11x^2y - xy^2 + y^3
Distribute the subtraction negative sign to all terms in the cost expression and group like terms together to obtain the final simplified expression.

Key Concept

Simplifying multi-variable algebraic expressions by distributing coefficients and combining like terms.
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Topics

Simplifying Expressions and Combining Like Terms

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