To demonstrate the Closure Property of multiplication of polynomials, we will find the product of the two given polynomials:
\((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2).\)
We can distribute the terms as follows:
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Multiply \(\frac{1}{2}x^2\) by each term in the second polynomial: \[ \frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3, \] \[ \frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4. \]
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Multiply \(-3\) by each term in the second polynomial: \[ -3 \cdot 4y^3 = -12y^3, \] \[ -3 \cdot 5x^2 = -15x^2. \]
Now, combine all the results: \[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2. \]
Thus, the product is: \[ \frac{5}{2}x^4 + 2x^2y^3 - 12y^3 - 15x^2. \]
Now, we can review the options provided to find the matching answer:
- A. \(\frac{5}{2}x^4 + 2x^2y^3 - 12y^3 - 15x^2\) (This is correct)
- B. \(x^2y^3 + \frac{5}{x^4} - 8y^3 - 15x^2\) (Incorrect, since it incorrectly includes a division term)
- C. \(2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2\) (This is also correct, but the first term \(2x^2y^3\) appears odd compared to the usual expansion)
- D. \(\sqrt{x^2y^3} + 4x^4 - 15x^2\) (Incorrect, since it includes an erroneous square root and wrong coefficients)
Choice A matches our expansion accurately.
Thus, the final answer is A.
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