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

Question

(12x2−3)(4y3+5x2)


(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

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

Answer 1

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

\[ (12x^2 - 3)(4y^3 + 5x^2) \]

Using the distributive property (FOIL method for binomials):

1. First: \(12x^2 \cdot 4y^3 = 48x^2y^3\)
2. Outer: \(12x^2 \cdot 5x^2 = 60x^4\)
3. Inner: \(-3 \cdot 4y^3 = -12y^3\)
4. Last: \(-3 \cdot 5x^2 = -15x^2\)

Now, combine all these results:

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

In standard form, we can rearrange:

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

Thus, the final polynomial is \(60x^4 + 48x^2y^3 - 12y^3 - 15x^2\).

None of the provided responses match this outcome, and all of them appear incorrect based on the multiplication performed. Thus, to satisfy the requirement of showing the Closure Property of multiplication of polynomials, the product is:

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

If you need to match it with one of the options provided, you may need to verify the calculations or check if the options contain a typo or error.