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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

5/2 x^2y^3+2x^4−8y^3+15x^2

x^2y^3+5/x^4 −8y^3−15x^2

2x^2y^3+5/2 x^4−12y^3−15x^2

\sqrt(x^(2)) y^3 + 4x^4 - 15x^2

3 answers

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

Let's perform the multiplication step-by-step:

  1. Distribute \( 12x^2 \) to each term in \( (4y^3 + 5x^2) \):

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

  2. Distribute \( -3 \) to each term in \( (4y^3 + 5x^2) \):

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

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

Rearranging the terms, we have: \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]

Now let's look at the provided responses:

  1. \( 5/2 x^2y^3 + 2x^4 - 8y^3 + 15x^2 \)
  2. \( x^2y^3 + 5/x^4 - 8y^3 - 15x^2 \)
  3. \( 2x^2y^3 + 5/2 x^4 - 12y^3 - 15x^2 \)
  4. \( \sqrt{x^{2}} y^3 + 4x^4 - 15x^2 \)

None of the answers appear to match the correct product we calculated, which is \( 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \).

Since none of the provided options reflect this correct multiplication of the original polynomials clearly, it could indicate a mistake in the options. But to ensure an understanding of the Closure Property of multiplication in polynomials, we can conclude that the product indeed remains a polynomial.

If I missed any specific instructions regarding selecting an answer or providing more context, please let me know!

just choose a, b, c, or d

Based on your request, here are the options labeled:

a. \( 5/2 x^2y^3 + 2x^4 - 8y^3 + 15x^2 \)

b. \( x^2y^3 + 5/x^4 - 8y^3 - 15x^2 \)

c. \( 2x^2y^3 + 5/2 x^4 - 12y^3 - 15x^2 \)

d. \( \sqrt{x^{2}} y^3 + 4x^4 - 15x^2 \)

Among the options, none accurately represent the product of \((12x^2 - 3)(4y^3 + 5x^2)\). However, considering the format and contents of the responses:

  • The correct structure of terms suggests option (a) is the best choice, as it contains polynomial terms and resembles the structure of a polynomial.

Thus, the answer is (a).