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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

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

x2y3+5x4−8y3−15x2
x squared y cubed plus Start Fraction 5 over x superscript 4 baseline End Fraction minus 8 y cubed minus 15 x squared

52x2y3+2x4−8y3+15x2

1 answer

To find the product of the polynomials \((12x^2 - 3)(4y^3 + 5x^2)\), we can use the distributive property (also known as the FOIL method for binomials). Here’s how to multiply the two polynomials step-by-step:

  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 these results:

    • \(48x^2y^3 + 60x^4 - 12y^3 - 15x^2\)

  4. Rearranging in standard form (generally, we put the highest degree term first):

    • \(60x^4 + 48x^2y^3 - 12y^3 - 15x^2\)

Now, looking at the provided responses, the terms we calculated match up with the second response:

Final Product: \(60x^4 + 48x^2y^3 - 12y^3 - 15x^2\)

Closure Property: This demonstrates the closure property of multiplication for polynomials, as the result of multiplying two polynomials is also a polynomial.

Thus, the correct answer from the options provided is: 2x^2y^3 + 52x^4 - 12y^3 - 15x^2. However, there seems to be a mix-up in terms of coefficients in that response. Please check to ensure the numerical values are accurate in the choices presented, as it may be a typographical error.