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

A) 2x^2y^3+5/2x^4−12y^3−15x^2
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

B) 5/2x^2y^3+2x^4−8y^3+15x^2
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

C) x^2y^3+5/x^4−8y^3−15x^2
x squared y cubed plus Start Fraction 5 over x superscript 4 baseline End Fraction minus 8 y cubed minus 15 x squared

D) √2^2y^3+4x^4−15x^2
Start Root x squared End Root y cubed plus 4 x superscript 4 baseline minus 15 x squared

Answer 1

To find the product of the polynomials \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\), we can use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial.

Let's break this down step-by-step:

1. Distribute \(\frac{1}{2}x^2\):

   • \(\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3\)
   • \(\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4\)

2. Distribute \(-3\):

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

Now, combine all these results together:

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

Now, arranging these terms gives us:

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

Looking at the provided responses, the correct answer is:

A) \(2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2\).