Question

Which of the following equations demonstrate that the set of polynomials is not closed under the certain operations?

A. Division: (x^2+2x) / (x+1) = x+ x/x+1
B. Multip.: (3x^4+x^3)(-2x^4+x^3)= -6x^6+ x^7+x^6
C. Addition: (3x^4+x^3)+(-2x^4+x^3)= x^4 + 2x^3
D. Multip.: (x^2+2x)(x+1)= x^3 +3x^2+2x

Answer 1

A. Division

`=(x^2+2 x)/(x+1)\\\\=(x(x+1)+x)/(x+1)\\\\=(x(x+1))/(x+1)+(x)/(x+1)\\\\= x+(x)/(x+1)`

→→→True

B. Multiplication

L HS

`(3 x^4+x^3)(-2 x^4+x^3)=3 x^4*(-2 x^4+x^3)+x^3 * (-2 x^4+x^3)\\\\=3 * -2* x^4 * x^4 +3 * 1* x^4 * x^3 + x^3 * x^4 * -2+x^3 * x^3\\\\=-6 x^8+3 x^7-2 x^7+x^6\\\\=-6 x^8+ x^7+x^6`

R HS

`=-6x^6+ x^7+x^6`

LHS ≠ RHS

→→→→False

C: Addition

L HS

`=(3x^4+x^3)+(-2x^4+x^3)\\\\=3x^4-2x^4+x^3+x^3\\\\ x^4+2 x^3=RHS`

→→→True

D: Multiplication

LHS

`=(x^2+2 x)(x+1)\\\\= x^2*(x+1)+2 x* (x+1)\\\\= x^3+x^2+2 x^2+2 x\\\\= x^3+3x^2+2 x=RHS`

→→→True

Option B: is not correctly closed under Multiplication.

Answer 2

Division: (x^2+2x) / (x+1) = x+ x/x+1