Question

Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials.
(1/2 x^2 – 3)(4y^3 + 5x^2)

Answer 1

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

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

We will use the distributive property (also known as the FOIL method for binomials) to perform this multiplication:

1. **Multiply each term in the first polynomial by each term in the second polynomial.**

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

2. **Now calculate each of these products:**

- First term:
\[
(1/2 x^2)(4y^3) = 2x^2y^3
\]

- Second term:
\[
(1/2 x^2)(5x^2) = \frac{5}{2} x^4
\]

- Third term:
\[
(-3)(4y^3) = -12y^3
\]

- Fourth term:
\[
(-3)(5x^2) = -15x^2
\]

3. **Now combine all the computed terms:**
\[
2x^2y^3 + \frac{5}{2} x^4 - 12y^3 - 15x^2
\]

4. **Rewrite the expression in standard polynomial form (i.e., arrange by decreasing degree):**
\[
\frac{5}{2} x^4 + 2x^2y^3 - 15x^2 - 12y^3
\]

From this multiplication, we can see that the product itself is a polynomial, which demonstrates the Closure Property of multiplication of polynomials. The polynomial remains within the set of polynomials after multiplication.

Thus, the final result of multiplying the two polynomials is:
\[
\frac{5}{2} x^4 + 2x^2y^3 - 15x^2 - 12y^3
\]