To demonstrate the Closure Property of multiplication of polynomials using the expression \((12x^2−3)(4y^3+5x^2)\), we need to multiply the two polynomials.
Let's perform the multiplication:
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Distribute \(12x^2\) to both terms in \( (4y^3 + 5x^2) \): \[ 12x^2 \cdot 4y^3 = 48x^2y^3 \] \[ 12x^2 \cdot 5x^2 = 60x^4 \]
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Distribute \(-3\) to both terms in \( (4y^3 + 5x^2) \): \[ -3 \cdot 4y^3 = -12y^3 \] \[ -3 \cdot 5x^2 = -15x^2 \]
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Now, combine all the products: \[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]
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Rearranging the terms gives us: \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]
Thus, the product is: \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]
This result is a polynomial, which shows that the multiplication of two polynomials yields another polynomial, demonstrating the Closure Property of multiplication of polynomials.
Based on the responses you provided, none of them match the resulting polynomial. However, the key takeaway is that the product is a polynomial and confirms the Closure Property. If you need shortcuts or alternative ways to represent it among the given responses, it appears they do not correctly depict the closure result.
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