To find the product of the polynomial \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\) and illustrate the closure property of multiplication of polynomials, we will multiply these two polynomials.
Step 1: Distribute each term in the first polynomial with each term in the second polynomial.
\[ \left(\frac{1}{2}x^2 - 3\right)(4y^3 + 5x^2) \]
Step 2: Multiply \(\frac{1}{2}x^2\) by each term in \(4y^3 + 5x^2\):
- \(\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3\)
- \(\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4\)
Step 3: Multiply \(-3\) by each term in \(4y^3 + 5x^2\):
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)
Step 4: Combine all results:
Now we combine all these products:
\[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
Step 5: Organizing the polynomial:
Putting it together, we write the final polynomial as:
\[ \frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3 \]
Conclusion:
The result \(\frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3\) is indeed a polynomial, which demonstrates the closure property of multiplication of polynomials: when two polynomials are multiplied, the result is also a polynomial.
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