To demonstrate the Closure Property of multiplication of polynomials, we need to multiply the two given polynomials: \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\).
Step 1: Use the distributive property (FOIL method) to multiply the polynomials.
We will multiply each term in the first polynomial by each term in the second polynomial.
- \(\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3\)
- \(\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4\)
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)
Step 2: Combine all the results.
Now, we gather all the products together:
\[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
Step 3: Rearrange the terms by degree.
For clarity, we can rearrange the terms in standard polynomial form:
\[ \frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3 \]
Conclusion:
The result of the multiplication \( \frac{1}{2}x^2 - 3 \) and \( 4y^3 + 5x^2 \) is:
\[ \frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3 \]
Since the result is also a polynomial, we can conclude that the multiplication of these two polynomials demonstrates the Closure Property of multiplication of polynomials.
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