To demonstrate that multiplying polynomials is a closed system, we'll multiply the polynomials (x2−1)(−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.