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

Question

(12x2−3)(4y3+5x2)

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

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

Answer 1

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 use the distributive property (also known as the FOIL method for binomials) to multiply these two polynomials.

1. First, multiply the first term of the first polynomial by each term of the second polynomial:

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

2. Next, multiply the second term of the first polynomial by each term of the second polynomial:

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

Now, we combine all of these products together:

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

Finally, we can rearrange the polynomial in standard form, which orders the terms by the degree of \(x\):

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

Since the result is a polynomial, we see that multiplication yields another polynomial, demonstrating the Closure Property of multiplication of polynomials.

Thus, none of the provided options match our calculated result directly. However, the important takeaway is that the product of two polynomials remains a polynomial, which is the essence of the Closure Property.