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

(12x2−3)(4y3+5x2)

(1 point)
Responses

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
Start Root x squared End Root y cubed plus 4 x superscript 4 baseline minus 15 x squared

x2y3+5x4−8y3−15x2

1 answer

To evaluate the product \((12x^2 - 3)(4y^3 + 5x^2)\) and demonstrate the Closure Property of multiplication of polynomials, we'll distribute each term in the first polynomial by each term in the second polynomial.

  1. Distributing \(12x^2\):

    • \(12x^2 \cdot 4y^3 = 48x^2y^3\)
    • \(12x^2 \cdot 5x^2 = 60x^4\)

  2. Distributing \(-3\):

    • \(-3 \cdot 4y^3 = -12y^3\)
    • \(-3 \cdot 5x^2 = -15x^2\)

Now, combine all the resulting terms:

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

Now, we can arrange the product in standard polynomial form (from highest degree term to lowest):

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

Comparing this with the given options, it appears none of the responses directly match the outcome. However, if we consider rearranging the terms based on their degrees or grouping similarly, we might see respondent similarities.

To finalize, none of the choices correctly represent our expanded form, but ensure no property violations as policies are closed under multiplication. The Closure Property is maintained since the result of multiplying two polynomials (another polynomial) isn’t void.

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