Question

Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials. (12x2−3)(4y3+5x2) (1 point) Responses 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 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

Answer 1

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

Using the distributive property (also known as the FOIL method for binomials), we can find the product:

1. Distributing \( 12x^2 \):

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

2. Distributing \( -3 \):

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

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

Now, if we rearrange this, we have: \[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]

As we see, the product \( (12x^2 - 3)(4y^3 + 5x^2) \) results in a polynomial, and therefore we demonstrate that the multiplication of two polynomials results in a polynomial, validating the Closure Property.

Thus, the result we obtained which shows the Closure Property is: \[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]

Among the options you provided, while this exact expression may not be listed, the essence of the multiplication showcases that the resultant expression remains a polynomial, affirming the Closure Property.

Answer 2

which answer is it

Answer 3

Let's examine the options you provided again in light of the correct product derived from multiplying the polynomials:

When we multiplied \( (12x^2 - 3) \) and \( (4y^3 + 5x^2) \), we found the product to be: \[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]

Now, let's compare this with the choices provided:

1. \( x^2y^3 + \frac{5}{x^4} - 8y^3 - 15x^2 \)
2. \( \frac{5}{2} x^2y^3 + 2x^4 - 8y^3 + 15x^2 \)
3. \( \sqrt{x^2}y^3 + 4x^4 - 15x^2 \)
4. \( 2x^2y^3 + 52x^4 - 12y^3 - 15x^2 \)

None of the answers match the correct product \( 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \) exactly, but let's approximate and analyze which one aligns closest based on coefficients:

• Option 1 contains a fractional term and incorrect coefficients.
• Option 2 also contains fractions and does not match coefficients.
• Option 3 has a radical and does not correlate.
• Option 4 has the term \( 2x^2y^3 \) and an inappropriate coefficient for \( x^4 \) but the structure resembles polynomial multiplication.

However, the best candidate seems to be Option 4. It is likely intended to represent a sum of terms which is also a polynomial.

Thus, while it does not provide an exact match, based on polynomial behavior, it appears that the closest or intended answer to demonstrate closure would be:

2x²y³ + 52x⁴ - 12y³ - 15x²

If my earlier calculations were intended to align more clearly with one of your options, please provide explicit variable powers or any other details that might impact the choice!