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

(1/2x^2 − 3)(4y^3 + 5x^2)

(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
x squared y cubed plus Start Fraction 5 over x superscript 4 baseline End Fraction minus 8 y cubed minus 15 x squared

x2−−√y3+4x4−15x2

1 answer

To find the product of the polynomials \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\) and demonstrate the Closure Property of multiplication, we will distribute each term in the first polynomial by each term in the second polynomial.

Here are the steps:

  1. Distribute \(\frac{1}{2}x^2\): \[ \frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3 \] \[ \frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4 \]

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

  3. Combine all the results: \[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]

So the final product is: \[ \frac{5}{2}x^4 + 2x^2y^3 - 12y^3 - 15x^2 \]

Among the options provided, the one that matches this result is: \[ \frac{5}{2}x^4 + 2x^2y^3 - 12y^3 - 15x^2 \]

Thus, this illustrates the Closure Property of multiplication of polynomials, as the result is also a polynomial.