To demonstrate the Closure Property of multiplication of polynomials using the given polynomials \((12x^2 - 3)\) and \((4y^3 + 5x^2)\), we will multiply them together:
\[ (12x^2 - 3)(4y^3 + 5x^2) \]
Using the distributive property (also known as the FOIL method for binomials), we multiply each term in the first polynomial by each term in the second polynomial:
- \(12x^2 \cdot 4y^3 = 48x^2y^3\)
- \(12x^2 \cdot 5x^2 = 60x^4\)
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)
Now combining all these results, we have:
\[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]
This shows that the product is also a polynomial, which demonstrates the closure property of multiplication of polynomials.
If we were to arrange this polynomial by descending powers of \(x\), we have:
\[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]
However, it seems there is no explicitly listed option from your choices matching exactly what we've computed. Let's look closer at the choices:
- Choice Analysis:
- \(52x^2y^3 + 2x^4 - 8y^3 + 15x^2\)
- \(x^2\sqrt{y^3} + 4x^4 - 15x^2\)
- \(2x^2y^3 + 52x^4 - 12y^3 - 15x^2\)
- \(x^2y^3 + 5x^4 - 8y^3 - 15x^2\)
None of these products match our expansion. Do double-check the polynomial we're multiplying with the given answers.
For completeness, just to ensure the closure property holds, our derived polynomial \(60x^4 + 48x^2y^3 - 15x^2 - 12y^3\) shows that the product of the two polynomials is also a polynomial, verifying the closure property of multiplication of polynomials.
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