Question

Use multiplication to find the product that demonstrates the closure property of the multiplication of polynomials. (1/2x^2 - 3) (4y^3+5x^2) A. 2x^2y^3 + 5/2x^4 - 12y^3 - 15x^2 B. 5/2 x^2y^3 + 2x^4 - 8y^3 + 15x^2 C. x^2y^3 + 5/x^4 - 8y^3 - 15x^2 D. √x^2y^3 + 4x^4 -15x^2

Answer 1

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:

1. Distribute \(\frac{1}{2}x^2\) across \((4y^3 + 5x^2)\):
\[
\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3
\]
\[
\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4
\]

2. Now distribute \(-3\) across \((4y^3 + 5x^2)\):
\[
-3 \cdot 4y^3 = -12y^3
\]
\[
-3 \cdot 5x^2 = -15x^2
\]

3. 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.