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

Question

(12x2−3)(4y3+5x2)

(1 point)
Responses

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
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

2x2y3+52x4−12y3−15x2

Answer 1

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

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

1. Distribute each term in the first polynomial to each term in the second polynomial: \[ (12x^2)(4y^3) + (12x^2)(5x^2) + (-3)(4y^3) + (-3)(5x^2) \]

2. Now, calculate each of these products:

   • \(12x^2 \cdot 4y^3 = 48x^2y^3\)
   • \(12x^2 \cdot 5x^2 = 60x^4\)
   • \(-3 \cdot 4y^3 = -12y^3\)
   • \(-3 \cdot 5x^2 = -15x^2\)

3. Collecting all these terms together gives: \[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]

4. Rearranging the terms, we can express it as: \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]

Now looking at the options:

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

None of the provided responses seem to correspond directly to the simplified polynomial we derived through multiplication.

However, based on the Closure Property, we can conclude that multiplying two polynomials yields another polynomial, which we have just demonstrated.

Thus, the product \((12x^2 - 3)(4y^3 + 5x^2)\) produces a polynomial: \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]

This demonstrates the Closure Property of multiplication of polynomials because the result is a polynomial. If you are looking for one of the exact response options provided, please double-check to see if there's a mismatch or typo, as none are accurate according to the calculation here.

Answer 2

you only looked that 3 options, i gave you 4

Answer 3

I apologize for overlooking one of the options! Let's take another look at all four options you've provided and compare them against the product we calculated.

We calculated the product of the polynomials \((12x^2 - 3)(4y^3 + 5x^2)\) and obtained:

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

Now, let's review all four options you provided:

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

None of these options seem to match the simplified polynomial \(60x^4 + 48x^2y^3 - 15x^2 - 12y^3\) that we derived.

To be clear, let's break down the product of the polynomials one more time:

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

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

2. Distributing \( -3 \):

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

Combining all the products, we again find: \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]

None of the options provided correspond to this result.

This confirms that the exercise demonstrates the Closure Property of multiplication of polynomials, as we have produced a polynomial from multiplying two polynomials. However, it seems the options may either contain a typographical error or do not directly reflect the intended answer.

If you're seeking the closest match or a specific aspect of the computation, let me know! Otherwise, we can conclude that none of the provided options accurately reflect the result of the polynomial multiplication given.