To demonstrate the closure property of multiplication for polynomials, we need to multiply the two given polynomials \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\).
Let's perform the multiplication step by step:
-
Distribute \(\frac{1}{2}x^2\) across \((4y^3 + 5x^2)\): \[ \frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3 \] \[ \frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4 \]
-
Now distribute \(-3\) across \((4y^3 + 5x^2)\): \[ -3 \cdot 4y^3 = -12y^3 \] \[ -3 \cdot 5x^2 = -15x^2 \]
-
Now, combine all the results: \[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
Now we can match this result with the options provided:
A. \(2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2\)
B. \(\frac{5}{2}x^2y^3 + 2x^4 - 8y^3 + 15x^2\)
C. \(x^2y^3 + \frac{5}{x^4} - 8y^3 - 15x^2\)
D. \(\sqrt{x^2y^3} + 4x^4 - 15x^2\)
The correct matching result is option A: \[ \text{A. } 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
This demonstrates the closure property of the multiplication of polynomials, as the result is also a polynomial.