1) Polynomials are closed under multiplication. What does that mean? When you multiply polynomials, the result is always a polynomial. When you multiply polynomials, the result is sometimes not a polynomial. When you multiply polynomials, the result is never a polynomial. Whe

The statement "Polynomials are closed under multiplication" means that when you multiply two polynomials together, the result is always a polynomial. To clarify: 1. **Polynomials** are expressions that consist of variables raised to non-negative integer powers and coefficients. For example, \( P(x) = 2x^2 + 3x + 1 \) and \( Q(x) = x + 4 \) are both polynomials. 2. **Closure under multiplication** means that if you take any two polynomials and multiply them, the product will also be a polynomial. For example: - If you multiply \( P(x) = 2x^2 + 3x + 1 \) and \( Q(x) = x + 4 \): \[ P(x) \cdot Q(x) = (2x^2 + 3x + 1)(x + 4) \] When you expand this product, you will get: \[ = 2x^3 + 8x^2 + 3x^2 + 12x + 1x + 4 \] \[ = 2x^3 + 11x^2 + 12x + 4 \] This result, \( 2x^3 + 11x^2 + 12x + 4 \), is also a polynomial. Thus, the correct interpretation of the statement is: - **When you multiply polynomials, the result is always a polynomial.**