Question

Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials.

(1/2x^2−3)(4y^3+5x^2)

Answer 1

To demonstrate the Closure Property of multiplication of polynomials, we need to multiply the two given polynomials: \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\).

**Step 1: Use the distributive property (FOIL method) to multiply the polynomials.**

We will multiply each term in the first polynomial by each term in the second polynomial.

1. \(\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3\)
2. \(\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4\)
3. \(-3 \cdot 4y^3 = -12y^3\)
4. \(-3 \cdot 5x^2 = -15x^2\)

**Step 2: Combine all the results.**

Now, we gather all the products together:

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

**Step 3: Rearrange the terms by degree.**

For clarity, we can rearrange the terms in standard polynomial form:

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

**Conclusion:**

The result of the multiplication \( \frac{1}{2}x^2 - 3 \) and \( 4y^3 + 5x^2 \) is:

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

Since the result is also a polynomial, we can conclude that the multiplication of these two polynomials demonstrates the Closure Property of multiplication of polynomials.