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:

Step1: Define closure property

Closure means result is in same set. For polynomials, if we add or subtract two polynomials, result is still a polynomial.

Step2: Addition example

Let \(P(x)=2x^2 + 3x+1\) and \(Q(x)=x^2 - 2x + 3\). Then \(P(x)+Q(x)=(2x^2 + 3x + 1)+(x^2-2x + 3)=3x^2+x + 4\), which is a polynomial.

Step3: Subtraction example

Using the same \(P(x)\) and \(Q(x)\), \(P(x)-Q(x)=(2x^2 + 3x + 1)-(x^2-2x + 3)=2x^2+3x + 1 - x^2 + 2x-3=x^2+5x - 2\), which is a polynomial.

Answer:

The statement "Polynomials are closed under the operations of addition and subtraction" means that when we add or subtract two polynomials, the resulting expression is also a polynomial. Addition example: If \(P(x)=2x^2 + 3x+1\) and \(Q(x)=x^2 - 2x + 3\), then \(P(x)+Q(x)=3x^2+x + 4\). Subtraction example: If \(P(x)=2x^2 + 3x + 1\) and \(Q(x)=x^2-2x + 3\), then \(P(x)-Q(x)=x^2+5x - 2\).