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 1

1. Polynomials are closed under the operations of addition and substraction mean that:

If we add two polynomials, the result is still a polynomial (so still in the set of polynomials)

Also, If we subtract 2 polynomials, the result is a polynomial.

2. Example: given the polynomials :

$M(x)= 3x^{2}+5x-1$ and $N(x)=-6 x^{4}+9 x^{3}+ x^{2} -2x-1$

Addition example: 

$M(x)+N(x)=(3x^{2}+5x-1)+(-6 x^{4}+9 x^{3}+ x^{2} -2x-1)=$

$-6 x^{4}+9 x^{3}+x^{2} +3x^{2}-2x+5x-1-1=-6 x^{4}+9 x^{3}+4x^{2}x-2$


Subtraction example: 

$N(x)-M(x)=(-6 x^{4}+9 x^{3}+ x^{2} -2x-1)-(3x^{2}+5x-1)$

=$-6 x^{4}+9 x^{3}+ x^{2} -2x-1-3x^{2}-5x+1=$
=$-6 x^{4}+9 x^{3}+ x^{2} -3x^{2}-2x-5x-1+1=-6 x^{4}+9 x^{3}-2x^{2}-x$

So both M(x)+N(x) and N(x)-M(x) are polynomials

Answer 2

After the addition and subtraction of any polynomial with the other polynomial the outcome is a polynomial. The below example is the prove of above statement.

The statement "Polynomial are closed under the operations of addition and subtraction" means that when the addition or subtraction of any polynomial with other polynomial is carried out the outcome is a polynomial.

Example: Add and subtract the polynomial given below-

$f(x) = x^2 + 2x +6$   and  $g(x) = 3x^2 +7x +16$

Addition: $f(x) + g(x) = x^2+2x +6 + 3x^2+7x +16=4x^2 + 9x +22$

Subtraction: $f(x)- g(x) = x^2 +2x +6-(3x^2+7x+16)=-2x^2 -5x -10$

So, from the above example it can be concluded that after the subtraction and addition of the polynomial the outcome is a polynomial.

For more information, refer the link given below:

https://brainly.com/question/23792383