Question

Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials. (1/2x^2 - 3) (4y^3 + 5x^2)
2x^2y^3 + 5/2x^4 - 12y^3 - 15x^2
5/2x^2y^3 +2x^4 - 8y^3 + 15x^2
√x^2y^3 + 4x^4 - 15x^2
x^2y^3 + 5/x^4 - 8y^3 - 15x^2

Answer 1

To demonstrate the Closure Property of multiplication of polynomials, we will multiply the two given polynomials:

\[(\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\]

We will perform the multiplication using the distributive property (also known as the FOIL method for binomials).

1. Distribute \(\frac{1}{2}x^2\) to both terms in the second polynomial:
- \(\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3\)
- \(\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4\)

2. Distribute \(-3\) to both terms in the second polynomial:
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)

3. Combine all the results:
\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2
\]

So the product of the two polynomials is:

\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2
\]

This demonstrates the Closure Property of multiplication of polynomials, as the result is also a polynomial.

Among the provided options, the correct result is:

**2x^2y^3 + 5/2x^4 - 12y^3 - 15x^2**