To demonstrate the Closure Property of multiplication of polynomials, we will multiply the given polynomials \( (12x^2 - 3) \) and \( (4y^3 + 5x^2) \).
Here's how we can do 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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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 the results: \[ (12x^2 - 3)(4y^3 + 5x^2) = 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]
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Now let's group similar terms: \[ 60x^4 + (48x^2y^3 - 15x^2 - 12y^3) \]
Thus, the final product of multiplying the polynomials \( (12x^2 - 3) \) and \( (4y^3 + 5x^2) \) is: \[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]
This result demonstrates the Closure Property of multiplication of polynomials because the product of two polynomials is also a polynomial.