Question

Use multiplication to find the product that demonstrates
the Closure Property of multiplication of polynomials.
(x² − 3) (4y³ + 5x²)
-
(1 point)
x²y³ +
5
x4
-
8y³ — 15x2
2x² y³ + 2x² - 12y³ – 15x²
-
2 3
√x²y³ + 4x4
15x2
x²y³ + 2x² − 8y³ + 15x²

Answer 1

To demonstrate the Closure Property of multiplication of polynomials, we need to find the product of the two given polynomials \( (x^2 - 3)(4y^3 + 5x^2) \).

Let's perform the multiplication step-by-step:

1. **Distribute \( x^2 \) to both terms in the second polynomial:**
\[
x^2 \cdot (4y^3 + 5x^2) = 4x^2y^3 + 5x^4
\]

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

3. **Combine all the results:**
\[
(x^2 - 3)(4y^3 + 5x^2) = 4x^2y^3 + 5x^4 - 12y^3 - 15x^2
\]

So, the final result is:
\[
5x^4 + 4x^2y^3 - 12y^3 - 15x^2
\]

This result is a polynomial, which shows that the product of two polynomials is still a polynomial, thereby demonstrating the Closure Property of multiplication for polynomials.