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Difficulty: HardMultiplying Polynomials
When the polynomial expression (3x−2y)(Ax2+4xy−5y2)(3x - 2y)(Ax^2 + 4xy - 5y^2) is fully expanded and simplified, the coefficient of the x2yx^2y term is −4-4. What is the value of the constant AA?
  1. A
    22
  2. 88Answer
  3. C
    −4-4
  4. D
    −8-8

Answer

The value of the constant AA is 88.
To find the coefficient of the x2yx^2y term, we do not need to expand the entire expression. We only need to find the specific multiplications that result in an x2yx^2y term. Multiplying the xx term from the first polynomial by the xyxy term from the second gives 3x×4xy=12x2y3x \times 4xy = 12x^2y. Multiplying the yy term from the first polynomial by the x2x^2 term from the second gives −2y×Ax2=−2Ax2y-2y \times Ax^2 = -2Ax^2y. Combining these gives a total x2yx^2y coefficient of (12−2A)(12 - 2A). Since the problem states this coefficient is −4-4, we set up the equation 12−2A=−412 - 2A = -4. Subtracting 1212 from both sides gives −2A=−16-2A = -16. Dividing by −2-2 gives A=8A = 8.

Step-by-Step Solution

1
Identify the products that result in an x2yx^2y term when expanding the polynomials.
The relevant terms are (3x)(4xy)(3x)(4xy) and (−2y)(Ax2)(-2y)(Ax^2).
Only these specific combinations of terms will produce the exact variable configuration x2yx^2y.
2
Multiply these specific terms to find their coefficients.
(3x)(4xy)=12x2y(3x)(4xy) = 12x^2y and (−2y)(Ax2)=−2Ax2y(-2y)(Ax^2) = -2Ax^2y.
This isolates the parts of the expansion that contribute to the final x2yx^2y coefficient.
3
Combine the coefficients and set them equal to the given total coefficient of −4-4.
12−2A=−412 - 2A = -4
The algebraic sum of the individual x2yx^2y coefficients must equal the final simplified coefficient provided in the problem.
4
Solve the linear equation for AA.
−2A=−4−12⇒−2A=−16⇒A=8-2A = -4 - 12 \Rightarrow -2A = -16 \Rightarrow A = 8
Standard algebraic isolation of the variable AA gives the final answer.

Key Concept

Identifying specific terms in polynomial multiplication and solving for an unknown coefficient.
Estimated Time:1m 30s
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Topics

Multiplying Polynomials

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