Question

find the error. your classmate says that polynomials are not closed under addition and gives this counterexample: ((x^2 - 2x) + (-x^2 + 2x) = 0). describe the error. just because our answer is 0, does that mean its not a polynomial? i can remind my classmate that (0x^2) and (0x) are both... because they are a product of a number, 0, and one or more variables. 0 is also a polynomial because it can be written as (0x^2 + 0x + 0). therefore, the solution is a...

Explanation:

Step1: Recall polynomial closure under addition

Polynomials are closed under addition, meaning the sum of two polynomials is a polynomial. A polynomial is an expression of the form \(a_nx^n + \dots + a_1x + a_0\), where \(a_i\) are constants and \(n\) is a non - negative integer. The zero polynomial (e.g., \(0x^2+0x + 0\)) is a polynomial (it can be written in the polynomial form with all coefficients zero).

Step2: Analyze the classmate's error

The classmate claims that \((x^2 - 2x)+(-x^2 + 2x)=0\) is a counterexample to the closure of polynomials under addition. But the sum \(0\) (which can be written as \(0x^2+0x + 0\)) is a polynomial. The error is in thinking that \(0\) is not a polynomial. In fact, the zero polynomial is a valid polynomial (it has degree \(-\infty\) or is considered as a polynomial with all coefficients zero). Also, when we simplify \((x^2 - 2x)+(-x^2 + 2x)\), we combine like terms: \(x^2 - x^2-2x + 2x=(1 - 1)x^2+(-2 + 2)x=0x^2+0x = 0\), and \(0\) is a polynomial.

Answer:

The error is that the classmate incorrectly believes that \(0\) (the result of the addition of the two polynomials) is not a polynomial. In reality, the zero polynomial (which is what \(0\) is in polynomial form, \(0x^2 + 0x+0\)) is a valid polynomial, and polynomials are closed under addition because the sum of two polynomials is always a polynomial (in this case, the zero polynomial).