Question

Use the following multiplication problem to show how multiplying polynomials is a closed system

(x^2-1)(-2x+5)

Answer 1

To show how multiplying polynomials is a closed system using the example of the polynomial multiplication \((x^2 - 1)(-2x + 5)\), we will follow these steps:

1. **Distribute Each Term**: We'll distribute each term in the first polynomial by each term in the second polynomial.

\[
(x^2 - 1)(-2x + 5)
\]

This can be expanded as follows:

\[
= x^2 \cdot (-2x) + x^2 \cdot 5 + (-1) \cdot (-2x) + (-1) \cdot 5
\]

2. **Perform the Multiplications**: Now, carry out each multiplication:

- \(x^2 \cdot (-2x) = -2x^3\)
- \(x^2 \cdot 5 = 5x^2\)
- \((-1) \cdot (-2x) = 2x\)
- \((-1) \cdot 5 = -5\)

3. **Combine the Results**: Now we add up all the results we obtained:

\[
-2x^3 + 5x^2 + 2x - 5
\]

4. **Final Result**: The final result of the multiplication is:

\[
-2x^3 + 5x^2 + 2x - 5
\]

### Explanation of Closure
A set of numbers is said to be closed under an operation if applying the operation to members of the set results in members of the same set.

In this case, the set of polynomials is closed under multiplication since the result of multiplying two polynomials (in this case, \((x^2 - 1)\) and \((-2x + 5)\)) is also a polynomial (\(-2x^3 + 5x^2 + 2x - 5\)).

The resulting polynomial is well-defined and retains the properties of polynomials, such as having whole number (or integer) exponents and being a finite sum of terms. Therefore, we conclude that the multiplication of polynomials is indeed a closed system.