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

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