Question

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

Answer 1

To demonstrate the Closure Property of multiplication for polynomials, we'll first multiply the polynomials \((\frac{1}{2}x^2 - 3)\) and \((4y^3 + 5x^2)\).

We will apply the distributive property:

1. Multiply \(\frac{1}{2}x^2\) by each term 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. Multiply \(-3\) by each term in the second polynomial:
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)

Now, combine all the results:

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

Thus, the product of the two polynomials is:
\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2
\]

From the options provided, the closest match is:
**2x^2 * y^3 + 5/2 * x^4 - 12y^3 - 15x^2**

This expression demonstrates the Closure Property of multiplication of polynomials because the result is also a polynomial.