Question

Use the following multiplication problem to show how multiplying polynomials is a closed system.
(x^2 – 1)(-2x + 5)

Answer 1

To demonstrate that multiplying polynomials is a closed system, we'll perform the multiplication of the given polynomials \( (x^2 - 1) \) and \( (-2x + 5) \).

We start by applying the distributive property, also known as the FOIL method in the case of binomials.

Here are the steps in detail:

1. **Distribute each term in the first polynomial to each term in the second polynomial**:

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

2. **Perform the multiplications**:

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

3. **Combine all the results**:

Now we can combine all the terms:

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

4. **Write the final expression**:

So, the product of \( (x^2 - 1)(-2x + 5) \) is:

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

### Conclusion:

The resulting polynomial \( -2x^3 + 5x^2 + 2x - 5 \) is still a polynomial.

This demonstrates that when we multiply two polynomials, the result is also a polynomial, indicating that the set of polynomials is closed under multiplication.