Question

Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials. \left(\frac{1}{2}x^2-3\right)\left(4y^3+5x^2\right) (1 point) Responses \sqrt{x^2}y^3+4x^4-15x^2 Start Root x squared End Root y cubed plus 4 x superscript 4 baseline minus 15 x squared x^2y^3+\frac{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 2x^2y^3+\frac{5}{2}x^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 \frac{5}{2}x^2y^3+2x^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 will use the distributive property (also known as the FOIL method for binomials) to multiply these polynomials:

1. Multiply \(\frac{1}{2}x^2\) by each term 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. Multiply \(-3\) by each term in the second polynomial:
\[
-3 \cdot 4y^3 = -12y^3,
\]
\[
-3 \cdot 5x^2 = -15x^2.
\]

Now we can combine all these results:

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

This product is a polynomial since it is expressed as a finite sum of terms with non-negative integer exponents of \(x\) and \(y\).

Among the options given, the correct product that matches what we computed is:

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

So the response demonstrating the Closure Property of multiplication of polynomials is:

**2x²y³ + \(\frac{5}{2}\)x⁴ - 12y³ - 15x².**