Question

Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials. (12x2−3)(4y3+5x2) (1 point) Responses x2−−√y3+4x4−15x2 Start Root x squared End Root y cubed plus 4 x superscript 4 baseline 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 x2y3+5x4−8y3−15x2

Answer 1

To find the product of the polynomials \( (12x^2 - 3)(4y^3 + 5x^2) \), we will distribute each term in the first polynomial with each term in the second polynomial. This process is known as the distributive property.

Let's break it down step by step:

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

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

Now, combine all the terms:
\[
48x^2y^3 + 60x^4 - 12y^3 - 15x^2
\]

To rearrange the terms in standard form (descending order of powers of \(x\) and \(y\)):
\[
60x^4 + 48x^2y^3 - 12y^3 - 15x^2
\]

So the product that demonstrates the closure property of multiplication of polynomials is:
\[
60x^4 + 48x^2y^3 - 12y^3 - 15x^2
\]

None of the provided responses directly match this answer. If you are looking for a specific response among the options given, they may contain errors or may not be formatted correctly, but the above calculations show the correct result from the multiplication of the polynomials.