Question

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

(12x2−3)(4y3+5x2)


5/2x2y3+2x4−8y3+15x2

√y3+4x4−15x2

2x2y3+52x4−12y3−15x2

x2y3+5x4−8y3−15x2

Answer 1

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

Let's perform the multiplication:

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

We will use the distributive property (often referred to as the FOIL method for binomials) to expand the expression:

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, we combine all these results:

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

Rearranging this to standard form (typically in descending order of degree for each term):

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

Now that we've computed the product, we can note that the final expression \(60x^4 + 48x^2y^3 - 12y^3 - 15x^2\) is indeed a polynomial, demonstrating the Closure Property of multiplication of polynomials; the product of two polynomials is still a polynomial.

Now, comparing this with the options provided, it seems none of the options matches the resulting polynomial we calculated. Please double-check if there are any typographical errors in the choices or if there are other requirements.