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Difficulty: Very hardMultiplying Polynomials

Simplify the following polynomial expression completely:

(3x−2y)(2x2−xy+4y2)−x(x−3y)2(3x - 2y)(2x^2 - xy + 4y^2) - x(x - 3y)^2

Which of the following represents the fully expanded and simplified result?

  1. 5x3−x2y+5xy2−8y35x^3 - x^2y + 5xy^2 - 8y^3Answer
  2. B
    5x3−13x2y+23xy2−8y35x^3 - 13x^2y + 23xy^2 - 8y^3
  3. C
    5x3−7x2y+23xy2−8y35x^3 - 7x^2y + 23xy^2 - 8y^3
  4. D
    7x3−13x2y+23xy2−8y37x^3 - 13x^2y + 23xy^2 - 8y^3

Answer

The completely simplified expression is 5x3−x2y+5xy2−8y35x^3 - x^2y + 5xy^2 - 8y^3.
The correct answer is derived by carefully following the order of operations: first expanding both the left (3x−2y)(2x2−xy+4y2)(3x - 2y)(2x^2 - xy + 4y^2) and right x(x−3y)2x(x - 3y)^2 sides of the subtraction independently, and then combining the resulting polynomials. Distributing the negative sign correctly during the final subtraction yields 5x3−x2y+5xy2−8y35x^3 - x^2y + 5xy^2 - 8y^3.

Step-by-Step Solution

1
Expand the product of the binomial and the trinomial: (3x−2y)(2x2−xy+4y2)(3x - 2y)(2x^2 - xy + 4y^2).
6x3−3x2y+12xy2−4x2y+2xy2−8y36x^3 - 3x^2y + 12xy^2 - 4x^2y + 2xy^2 - 8y^3
Every term in the first binomial must be multiplied by every term in the trinomial to ensure full expansion.
2
Combine the like terms in the first expanded expression.
6x3−7x2y+14xy2−8y36x^3 - 7x^2y + 14xy^2 - 8y^3
Combining like terms simplifies the polynomial before performing the final subtraction operation.
3
Expand the squared binomial in the second part of the expression: (x−3y)2(x - 3y)^2.
x2−6xy+9y2x^2 - 6xy + 9y^2
A perfect square binomial (a−b)2(a-b)^2 expands to a2−2ab+b2a^2 - 2ab + b^2.
4
Distribute the xx into the expanded squared binomial: x(x2−6xy+9y2)x(x^2 - 6xy + 9y^2).
x3−6x2y+9xy2x^3 - 6x^2y + 9xy^2
The outer monomial must be distributed to each term in the trinomial.
5
Subtract the second result from the first result: (6x3−7x2y+14xy2−8y3)−(x3−6x2y+9xy2)(6x^3 - 7x^2y + 14xy^2 - 8y^3) - (x^3 - 6x^2y + 9xy^2).
6x3−7x2y+14xy2−8y3−x3+6x2y−9xy26x^3 - 7x^2y + 14xy^2 - 8y^3 - x^3 + 6x^2y - 9xy^2
Subtracting a polynomial requires distributing the negative sign to every term inside the parentheses.
6
Combine all remaining like terms.
5x3−x2y+5xy2−8y35x^3 - x^2y + 5xy^2 - 8y^3
Grouping x3x^3, x2yx^2y, and xy2xy^2 terms produces the final simplified answer in standard form.

Key Concept

Multiplying complex polynomials and applying subtraction across multiple terms
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Topics

Multiplying Polynomials

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