Question

explain what the following statement means:
polynomials are closed under the operations of addition and subtraction.
provide one addition example and one subtraction example to demonstrate.

Explanation:

Closure under addition/subtraction means when you add or subtract two polynomials, the result is always another polynomial (no non-polynomial terms like negative exponents or square roots of variables are created).

Addition Example:

Take two polynomials: $3x^2 + 2x + 1$ and $x^2 - 4x + 5$. Adding them combines like terms, which preserves the polynomial form.

Subtraction Example:

Take two polynomials: $5x^3 - 2x^2 + 7$ and $2x^3 + 3x - 1$. Subtracting them (distributing the negative sign and combining like terms) also preserves the polynomial form.

Answer:

The statement means that if you add or subtract any two polynomials, the resulting expression will also be a polynomial (no operations produce non-polynomial terms).

• Addition Example: $(3x^2 + 2x + 1) + (x^2 - 4x + 5) = 4x^2 - 2x + 6$ (the result is a polynomial)
• Subtraction Example: $(5x^3 - 2x^2 + 7) - (2x^3 + 3x - 1) = 3x^3 - 2x^2 - 3x + 8$ (the result is a polynomial)