Question

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

(12x2−3)(4y3+5x2)

(1 point)
Responses

x2y3+5x4−8y3−15x2
x squared y cubed plus Start Fraction 5 over x superscript 4 baseline End Fraction minus 8 y cubed minus 15 x squared

2x2y3+52x4−12y3−15x2
2 x squared y cubed plus Start Fraction 5 over 2 End Fraction x superscript 4 baseline minus 12 y cubed minus 15 x squared

52x2y3+2x4−8y3+15x2
Start Fraction 5 over 2 End Fraction x squared y cubed plus 2 x superscript 4 baseline minus 8 y cubed plus 15 x squared

x2−−√y3+4x4−15x2

Answer 1

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

\((12x^2 - 3)(4y^3 + 5x^2)\).

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

1. **Distribute \(12x^2\)**:
- \(12x^2 \cdot 4y^3 = 48x^2y^3\)
- \(12x^2 \cdot 5x^2 = 60x^4\)

2. **Distribute \(-3\)**:
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)

3. **Combine all the terms**:
- \(48x^2y^3 + 60x^4 - 12y^3 - 15x^2\)

Rearranging in standard form (usually in descending order with respect to the degrees of \(x\) and \(y\)) gives:
- \(60x^4 + 48x^2y^3 - 12y^3 - 15x^2\)

None of the provided responses exactly match this expression. However, we can compare terms, especially since they might have been simplified or rearranged.

Upon reviewing the responses:
- None of the provided options are correct based on the multiplication done, but in the context of demonstrating closure, we can still affirm that the product \(60x^4 + 48x^2y^3 - 12y^3 - 15x^2\) is indeed a polynomial and thus demonstrates the closure property for polynomials.

The correct expression that shows closure is \(60x^4 + 48x^2y^3 - 12y^3 - 15x^2\).