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.



ANSWER : If you add two polynomials, the sum is always a polynomial. Example: (2x2 + 3x) + (8x2 - 4x) = 10x2 - x If you subtract two polynomials, the difference is always a polynomial. Example: (2x2 + 3x) - (8x2 - 4x) = -6x2 + 7x

Answer 1

Answer: When adding two polynomials, the sum will always be a polynomial for example: (10x^2 + 6x) + (2x^2 + 12x) = 12x^2 + 18x. The same goes for subtracting the sum will also be a polynomial for example : (10x^2 + 6x) - (2x^2 + 12x) = 8x^2 - 6x.

Answer 2

To understand what this statement is suggesting about polynomials, we must first understand what is meant by "closed". A polynomial is a mathematical expression containing more than two algebraic terms. Polynomials are similar to integers in that they are also closed when it comes to addition and subtraction. What this means is that when undergoing addition or subtraction, the answer will be of the same form. Two integers added or subtracted will simply give another integer. The same is true for polynomials. Two polynomials added or subtracted will always give another polynomial.

You provided examples showing just that:

(2x² + 3x) + (8x² - 4x) = 10x² - x

Here two polynomials (the functions in parentheses) were added together and we simply resulted in another polynomial with the same algebraic terms of x² and x.

(2x² + 3x) - (8x² - 4x) = -6x² + 7x


Here two polynomials were subtracted and again resulted in a polynomial answer with the same algebraic terms.