Question

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

(12x2−3)(4y3+5x2)

52x2y3+2x4−8y3+15x2

x2−−√y3+4x4−15x2

2x2y3+52x4−12y3−15x2

x2y3+5x4−8y3−15x2

Answer 1

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

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

Using the distributive property (also known as the FOIL method for binomials), we multiply each term in the first polynomial by each term in the second polynomial:

1. \(12x^2 \cdot 4y^3 = 48x^2y^3\)
2. \(12x^2 \cdot 5x^2 = 60x^4\)
3. \(-3 \cdot 4y^3 = -12y^3\)
4. \(-3 \cdot 5x^2 = -15x^2\)

Now combining all these results, we have:

\[
48x^2y^3 + 60x^4 - 12y^3 - 15x^2
\]

This shows that the product is also a polynomial, which demonstrates the closure property of multiplication of polynomials.

If we were to arrange this polynomial by descending powers of \(x\), we have:

\[
60x^4 + 48x^2y^3 - 15x^2 - 12y^3
\]

However, it seems there is no explicitly listed option from your choices matching exactly what we've computed. Let's look closer at the choices:

- **Choice Analysis:**
1. \(52x^2y^3 + 2x^4 - 8y^3 + 15x^2\)
2. \(x^2\sqrt{y^3} + 4x^4 - 15x^2\)
3. \(2x^2y^3 + 52x^4 - 12y^3 - 15x^2\)
4. \(x^2y^3 + 5x^4 - 8y^3 - 15x^2\)

None of these products match our expansion. Do double-check the polynomial we're multiplying with the given answers.

For completeness, just to ensure the closure property holds, our derived polynomial \(60x^4 + 48x^2y^3 - 15x^2 - 12y^3\) shows that the product of the two polynomials is also a polynomial, verifying the closure property of multiplication of polynomials.