Question

the following statement means: polynomials are closed under the operations of addition and subtraction. give one addition example and one subtraction example to demonstrate. sample response: 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. compare your response to the sample response above. did your response ... ... explain what closure means for addition and give an example? ... explain what closure means for subtraction and give an example? done

Explanation:

Step1: Define closure for addition

Closure under addition for polynomials means sum is polynomial.

Step2: Provide addition example

\((2x^{2}+3x)+(8x^{2}-4x)=10x^{2}-x\)

Step3: Define closure for subtraction

Closure under subtraction for polynomials means difference is polynomial.

Step4: Provide subtraction example

\((2x^{2}+3x)-(8x^{2}-4x)= - 6x^{2}+7x\)

Answer:

Polynomials are closed under addition and subtraction. For addition, when we add two polynomials like \((2x^{2}+3x)\) and \((8x^{2}-4x)\), the sum \((2x^{2}+3x)+(8x^{2}-4x)=10x^{2}-x\) is a polynomial. For subtraction, when we subtract two polynomials such as \((2x^{2}+3x)-(8x^{2}-4x)= - 6x^{2}+7x\), the result is also a polynomial.