To demonstrate the Closure Property of multiplication of polynomials, we will multiply the two polynomials \( (1/2 x^2 - 3) \) and \( (4y^3 + 5x^2) \).
Let's compute the product step by step:
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Distribute \( (1/2 x^2) \) across \( (4y^3 + 5x^2) \): \[ (1/2 x^2)(4y^3) + (1/2 x^2)(5x^2) = 2x^2y^3 + \frac{5}{2}x^4 \]
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Distribute \( (-3) \) across \( (4y^3 + 5x^2) \): \[ -3(4y^3) - 3(5x^2) = -12y^3 - 15x^2 \]
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Combine all the terms from both distributions: \[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
Thus, the complete product can be written as: \[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
Now, let's compare this with the given options:
A) \( \sqrt{x^2y^3 + 4x^4 - 15^2} \)
B) \( 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \)
C) \( \frac{5}{2}x^2y^3 + 2x^4 - 8y^3 + 15x^2 \)
D) \( x^2y^3 + 2x^4 - 8y^3 - 15x^2 \)
The correct option is:
B) \( 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \).