To show that multiplying polynomials is a closed system, we can take the given polynomials \((x^2−1)\) and \((-2x+5)\) and perform the multiplication.
### Step 1: Distribute the terms
We apply the distributive property (also known as the FOIL method for binomials).
\[
(x^2 - 1)(-2x + 5)
\]
We can distribute each term in the first polynomial to each term in the second polynomial:
1. Multiply \(x^2\) by \(-2x\):
\[
x^2 \cdot -2x = -2x^3
\]
2. Multiply \(x^2\) by \(5\):
\[
x^2 \cdot 5 = 5x^2
\]
3. Multiply \(-1\) by \(-2x\):
\[
-1 \cdot -2x = 2x
\]
4. Multiply \(-1\) by \(5\):
\[
-1 \cdot 5 = -5
\]
### Step 2: Combine the results
Now, we combine all the terms together:
\[
-2x^3 + 5x^2 + 2x - 5
\]
### Step 3: Arrange the expression
We can arrange the final polynomial in standard form, which typically orders the terms from highest to lowest degree:
\[
-2x^3 + 5x^2 + 2x - 5
\]
### Conclusion
The result, \(-2x^3 + 5x^2 + 2x - 5\), is also a polynomial. This shows that the product of two polynomials (both of degree 2 and 1, respectively) is still a polynomial (in this case, degree 3).
Therefore, the multiplication of polynomials is a closed system, as the product of two polynomials is also a polynomial.