To demonstrate that multiplying polynomials is a closed system, we'll multiply the polynomials \( (x^2 - 1)(-2x + 5) \):

Question

1. **Distribute:**
Using the distributive property, we have:
\[
(x^2 - 1)(-2x + 5) = x^2 \cdot (-2x) + x^2 \cdot 5 - 1 \cdot (-2x) - 1 \cdot 5
\]

2. **Calculate each term:**
- \( x^2 \cdot (-2x) = -2x^3 \)
- \( x^2 \cdot 5 = 5x^2 \)
- \( -1 \cdot (-2x) = 2x \)
- \( -1 \cdot 5 = -5 \)

3. **Combine results:**
\[
-2x^3 + 5x^2 + 2x - 5
\]

4. **Conclusion:**
The result \( -2x^3 + 5x^2 + 2x - 5 \) is a polynomial, demonstrating that the set of polynomials is closed under multiplication.

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Answer 1

To demonstrate that the set of polynomials is closed under multiplication, we can multiply the polynomials \( (x^2 - 1)(-2x + 5) \):

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

2. Calculate each term:

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

3. Combine results: \[ -2x^3 + 5x^2 + 2x - 5 \]

4. Conclusion: The product \( -2x^3 + 5x^2 + 2x - 5 \) is a polynomial, showing that the set of polynomials is closed under multiplication.