To demonstrate the Closure Property of multiplication of polynomials, we need to multiply the two given polynomials: \((\frac{1}{2}x^2-3)(4y^3+5x^2)\).
Let's calculate the product step by step:
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Distribute \(\frac{1}{2}x^2\) to both terms 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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Next, distribute \(-3\) to both terms in the second polynomial: \[ -3 \cdot 4y^3 = -12y^3 \] \[ -3 \cdot 5x^2 = -15x^2 \]
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Now, combine all the results: \[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
So the final result after combining is: \[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
This matches one of your provided options: \[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
Thus, the product that demonstrates the Closure Property of multiplication of polynomials is: 2x²y³ + 5/2x⁴ - 12y³ - 15x²