Question

Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials. (12x2−3)(4y3+5x2) (1 point)

Answer 1

To demonstrate the Closure Property of multiplication concerning polynomials, we will multiply the two given polynomials: \( (12x^2 - 3) \) and \( (4y^3 + 5x^2) \).

Let's perform the multiplication step by step.

1. **Distribute \( 12x^2 \) to each term in the second polynomial:**
- \( 12x^2 \cdot 4y^3 = 48x^2y^3 \)
- \( 12x^2 \cdot 5x^2 = 60x^4 \)

2. **Now distribute \(-3\) to each term in the second polynomial:**
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)

3. **Combine all these results:**
- The expanded product is:
\[
48x^2y^3 + 60x^4 - 12y^3 - 15x^2
\]

4. **Now, combine like terms:**
- The terms \(48x^2y^3\) and \(-12y^3\) do not combine (as they are not like terms), and the \(60x^4\) stands alone.
- The terms \(60x^4\) and \(-15x^2\) also stand alone because they do not share the same degree.

Thus, the final product of the multiplication is:
\[
60x^4 + 48x^2y^3 - 12y^3 - 15x^2
\]

This polynomial, \( 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \), is still a polynomial, demonstrating the Closure Property of multiplication for polynomials.