Question

Are polynomial expressions closed under addition? Explain.

Answer 1

Yes, they are. A polynomial expression is defined as an expression with only whole number powers of any variable. So, for example, x^2+5x+6 is a polynomial, because 2, 1, and 0 are the powers of x, and those are all whole numbers. When you add and subtract variables with the same power, they just make that same variable to the same power with a new coefficient. The powers don't change, so you won't get any non-whole-number powers when you add and subtract polynomial expressions. That means that any two polynomial expressions added together is another polynomial, which is the definition of closure.