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

Question

(1/2x^2−3)(4y^3+5x^2)

(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 find the product of the two polynomials \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\), we will use the distributive property (also known as the FOIL method for binomials).

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

Thus, the final expression is: \[ \frac{5}{2}x^4 + 2x^2y^3 - 12y^3 - 15x^2 \]

This matches the answer selection of: \(2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2\)

So, the product that demonstrates the Closure Property of multiplication of polynomials is: \(2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2\).