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

Question

(1/2x2−3)(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

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

2x2y3+52x4−12y3−15x2

Answer 1

To demonstrate the Closure Property of multiplication of polynomials, we need to multiply the two given polynomials and simplify the expression:

The two polynomials are:

\[ (1/2x^2 - 3)(4y^3 + 5x^2) \]

Now we will apply the distributive property (also known as the FOIL method in the case of binomials). We will multiply each term in the first polynomial by each term in the second polynomial:

1. Multiply \( \frac{1}{2}x^2 \) by \( 4y^3 \): \[ \frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3 \]

2. Multiply \( \frac{1}{2}x^2 \) by \( 5x^2 \): \[ \frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4 \]

3. Multiply \( -3 \) by \( 4y^3 \): \[ -3 \cdot 4y^3 = -12y^3 \]

4. Multiply \( -3 \) by \( 5x^2 \): \[ -3 \cdot 5x^2 = -15x^2 \]

Now, we combine all the products:

\[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]

Now we can write the final result in a standard polynomial form:

\[ \frac{5}{2}x^4 + 2x^2y^3 - 12y^3 - 15x^2 \]

Among your provided options, the response that most closely matches this result is:

2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2

So the Closure Property of multiplication is demonstrated, and the multiplication of these two polynomials results in another polynomial.