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

Question

multiplication of polynomials.
$(\frac {1}{2}x^{2}-3)(4y^{3}+5x^{2})$
(1 point)
$\sqrt {x^{2}}y^{3}+4x^{4}-15x^{2}$
$\frac {5}{2}x^{2}y^{3}+2x^{4}-8y^{3}+15x^{2}$
$2x^{2}y^{3}+\frac {5}{2}x^{4}-12y^{3}-15x^{2}$
$x^{2}y^{3}+\frac {5}{x^{4}}-8y^{3}-15x^{2}$

Answer 1

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

\[ \left(\frac{1}{2}x^2 - 3\right)\left(4y^3 + 5x^2\right) \]

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

1. Distribute \(\frac{1}{2}x^2\) to both terms in the second polynomial: \[ \frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3 \] \[ \frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4 \]

2. Distribute \(-3\) to both terms in the second polynomial: \[ -3 \cdot 4y^3 = -12y^3 \] \[ -3 \cdot 5x^2 = -15x^2 \]

Now, we combine all the terms:

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

The fully multiplied polynomial is:

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

Now we can look for this expression among the provided answer choices. The correct answer is:

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