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Bot GPT 4-o mini To demonstrate the Closure Property of multiplication of polynomials, we need to multiply the two given polynomials: \( (12x^2 - 3)(4y^3 + 5x^2) \).
Let's perform the multiplication step-by-step:
1. Distribute \( 12x^2 \) to each term in \( (4y^3 + 5x^2) \):
- \( 12x^2 \cdot 4y^3 = 48x^2y^3 \)
- \( 12x^2 \cdot 5x^2 = 60x^4 \)
2. Distribute \( -3 \) to each term in \( (4y^3 + 5x^2) \):
- \( -3 \cdot 4y^3 = -12y^3 \)
- \( -3 \cdot 5x^2 = -15x^2 \)
Now, combine all these terms together:
\[
48x^2y^3 + 60x^4 - 12y^3 - 15x^2
\]
Rearranging the terms, we have:
\[
60x^4 + 48x^2y^3 - 15x^2 - 12y^3
\]
Now let's look at the provided responses:
1. \( 5/2 x^2y^3 + 2x^4 - 8y^3 + 15x^2 \)
2. \( x^2y^3 + 5/x^4 - 8y^3 - 15x^2 \)
3. \( 2x^2y^3 + 5/2 x^4 - 12y^3 - 15x^2 \)
4. \( \sqrt{x^{2}} y^3 + 4x^4 - 15x^2 \)
None of the answers appear to match the correct product we calculated, which is \( 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \).
Since none of the provided options reflect this correct multiplication of the original polynomials clearly, it could indicate a mistake in the options. But to ensure an understanding of the Closure Property of multiplication in polynomials, we can conclude that the product indeed remains a polynomial.
If I missed any specific instructions regarding selecting an answer or providing more context, please let me know!
anonymous
answered