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

Question

(12x2−3)(4y3+5x2)

(1 point)
Responses

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
Start Root x squared End Root y cubed plus 4 x superscript 4 baseline minus 15 x squared

52x2y3+2x4−8y3+15x2

Answer 1

To find the product of the polynomials \( (12x^2 - 3)(4y^3 + 5x^2) \), we will use the distributive property (also known as the FOIL method for binomials).

Here's the step-by-step multiplication:

1. Distribute \( 12x^2 \) to both terms in the second polynomial:

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

2. Distribute \( -3 \) to both terms in the second polynomial:

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

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

Next, we can write the resulting polynomial in standard form (usually ordered by degree): \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]

Now let's check which of the provided responses matches our result:

• The response that matches our result is: None of the provided options exactly match the resulting polynomial.

Therefore, for demonstration purposes, we have calculated the product of the multiplication, which confirms the closure property of multiplication of polynomials; the product is indeed a polynomial.

If there's a requirement for a choice from the options given when none match, I suggest reviewing the options for any potential typographical errors or misinterpretations in the polynomials presented.

Answer 2

Pick the one that is most right

Answer 3

Given the options provided and our calculated result of:

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

Let's see if any of the options closely resemble our polynomial:

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

The option \(2x^2y^3 + 52x^4 - 12y^3 - 15x^2\) is the closest, as it has similar terms but with a different coefficient for \(x^4\) and \(x^2y^3\).

However, I would suggest that it appears to use a factor of \(\frac{1}{26}\) or similar contextually, which makes it seem plausible as a potential misrepresentation of the correct polynomial result.

In this case, I would select:

\(2x^2y^3 + 52x^4 - 12y^3 - 15x^2\)

as the option that is "most right," though it's important to note it is still not an exact match to our calculated result.