To demonstrate the Closure Property of multiplication concerning polynomials, we will multiply the two given polynomials: \( (12x^2 - 3) \) and \( (4y^3 + 5x^2) \).
Let's perform the multiplication step by step.
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Distribute \( 12x^2 \) to each term in the second polynomial:
- \( 12x^2 \cdot 4y^3 = 48x^2y^3 \)
- \( 12x^2 \cdot 5x^2 = 60x^4 \)
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Now distribute \(-3\) to each term in the second polynomial:
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)
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Combine all these results:
- The expanded product is: \[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]
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Now, combine like terms:
- The terms \(48x^2y^3\) and \(-12y^3\) do not combine (as they are not like terms), and the \(60x^4\) stands alone.
- The terms \(60x^4\) and \(-15x^2\) also stand alone because they do not share the same degree.
Thus, the final product of the multiplication is: \[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]
This polynomial, \( 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \), is still a polynomial, demonstrating the Closure Property of multiplication for polynomials.