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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

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

1 answer

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

We'll multiply each term in the first polynomial by each term in the second polynomial.

  1. Multiply \(12x^2\) by \(4y^3\): \[ 12x^2 \cdot 4y^3 = 48x^2y^3 \]

  2. Multiply \(12x^2\) by \(5x^2\): \[ 12x^2 \cdot 5x^2 = 60x^4 \]

  3. Multiply \(-3\) by \(4y^3\): \[ -3 \cdot 4y^3 = -12y^3 \]

  4. Multiply \(-3\) by \(5x^2\): \[ -3 \cdot 5x^2 = -15x^2 \]

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

Thus, the product of the two polynomials is: \[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]

Now we can summarize:

  • The Closure Property of multiplication tells us that when we multiply two polynomials, the result is also a polynomial.

The final answer that demonstrates the Closure Property of multiplication of the given polynomials is: \[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]

However, none of the response options provided match our result. Please double-check the options for correctness.