To demonstrate the Closure Property of multiplication of polynomials using the given polynomials \( (12x^2 - 3) \) and \( (4y^3 + 5x^2) \), we need to multiply these two polynomials together.
We will use the distributive property (also known as the FOIL method for binomials) to perform the multiplication:
\[ (12x^2 - 3)(4y^3 + 5x^2) \]
Distributing:
- \( 12x^2 \cdot 4y^3 = 48x^2y^3 \)
- \( 12x^2 \cdot 5x^2 = 60x^4 \)
- \( -3 \cdot 4y^3 = -12y^3 \)
- \( -3 \cdot 5x^2 = -15x^2 \)
Now, we add these results together:
\[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]
Now we can reorder the terms in standard polynomial form (typically in descending order of degree):
\[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]
This product successfully demonstrates the Closure Property of multiplication of polynomials, as the result is another polynomial.
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