Question

To demonstrate that polynomials form a closed system under multiplication, we need to show that when two polynomials are multiplied, the result is also a polynomial.
1. **Identifying Polynomials**: Let's take some example polynomials:
- \( P(x) = x^4 - 2 \)
- \( Q(x) = 2x^3 + x^2 \)
- \( R(x) = 5x + 2 \)

2. **Multiplication of Polynomials**: If we take any two of the above polynomials and multiply them, we need to show that the result is still a polynomial.

Let's multiply \( P(x) \) and \( Q(x) \):
\[
P(x) \cdot Q(x) = (x^4 - 2)(2x^3 + x^2)
\]

Using the distributive property:
\[
= x^4 \cdot 2x^3 + x^4 \cdot x^2 - 2 \cdot 2x^3 - 2 \cdot x^2
\]
\[
= 2x^7 + x^6 - 4x^3 - 2x^2
\]

The result \( 2x^7 + x^6 - 4x^3 - 2x^2 \) is indeed a polynomial.

3. **Conclusion**: Thus, since multiplying two polynomials \( P(x) \) and \( Q(x) \) resulted in another polynomial, we confirm that polynomials are closed under multiplication.

If you have a specific set of polynomial expressions or numbers to evaluate in mind (perhaps derived from your question), please clarify, and I'd be glad to assist further!
MAKE IT SMALLER

Answer 1

To demonstrate that polynomials form a closed system under multiplication, we need to show that the product of two polynomials is also a polynomial.

1. **Example Polynomials**:
- \( P(x) = x^4 - 2 \)
- \( Q(x) = 2x^3 + x^2 \)

2. **Multiplication**:
Multiply \( P(x) \) and \( Q(x) \):
\[
P(x) \cdot Q(x) = (x^4 - 2)(2x^3 + x^2)
\]

Using the distributive property:
\[
= x^4 \cdot 2x^3 + x^4 \cdot x^2 - 2 \cdot 2x^3 - 2 \cdot x^2
\]
\[
= 2x^7 + x^6 - 4x^3 - 2x^2
\]

The result \( 2x^7 + x^6 - 4x^3 - 2x^2 \) is a polynomial.

3. **Conclusion**:
Thus, since the product of the polynomials \( P(x) \) and \( Q(x) \) is also a polynomial, we confirm that polynomials are closed under multiplication.