To demonstrate that the multiplication of polynomials is a closed system, let's multiply the two given polynomials, \((x^2 - 1)\) and \((-2x + 5)\).
### Step 1: Distribute Each Term
We will use the distributive property (also known as the FOIL method when dealing with binomials) to multiply the polynomials.
\[
(x^2 - 1)(-2x + 5) = x^2 \cdot (-2x) + x^2 \cdot 5 - 1 \cdot (-2x) - 1 \cdot 5
\]
### Step 2: Perform the Multiplication
Calculating each term separately:
1. \(x^2 \cdot (-2x) = -2x^3\)
2. \(x^2 \cdot 5 = 5x^2\)
3. \(-1 \cdot (-2x) = 2x\)
4. \(-1 \cdot 5 = -5\)
### Step 3: Combine Like Terms
Now, let's combine all these results:
\[
-2x^3 + 5x^2 + 2x - 5
\]
### Conclusion
The result of the multiplication \((-2x^3 + 5x^2 + 2x - 5)\) is a polynomial because it consists of terms where each term includes a coefficient multiplied by a variable raised to a nonnegative integer exponent (in this case, the exponents are 3, 2, 1, and 0).
### Verifying Closed System
Since we started with two polynomials and multiplied them, and the result is still a polynomial, we can conclude that the set of polynomials is closed under multiplication. Thus, multiplying polynomials results in another polynomial, demonstrating that polynomials indeed form a closed system under multiplication.