Question

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

(12x2−3)(4y3+5x2)

(1 point)
Responses

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

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

x2y3+5x4−8y3−15x2

Answer 1

To find the product of the polynomials \((12x^2 - 3)(4y^3 + 5x^2)\) using multiplication, we can apply the distributive property (also known as the FOIL method for binomials):

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

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

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

Rearranging this, we get:
- \(60x^4 + 48x^2y^3 - 12y^3 - 15x^2\)

So, we can write the product as:
\[
60x^4 + 48x^2y^3 - 15x^2 - 12y^3
\]

Now, looking at the provided options, the closest representation of our result would be a rearranged version of the polynomial. Based on the context given, the correct choice might not be explicitly listed, or it could be a combination of terms.

It's important to also remember that the Closure Property of multiplication for polynomials states that the product of two polynomials is also a polynomial. Since the product we computed is indeed a polynomial, it demonstrates the Closure Property.

If none of the options correctly represent the exact polynomial we calculated, please check again for any errors in transcription or selection.