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

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

Answer 2

Solution:

A. Division

  $=\frac{x^2+2 x}{x+1}\\\\=\frac{x(x+1)+x}{x+1}\\\\=\frac{x(x+1)}{x+1}+\frac{x}{x+1}\\\\= x+\frac{x}{x+1}$

→→→True

B. Multiplication

L HS

$(3 x^4+x^3)(-2 x^4+x^3)=3 x^4\times(-2 x^4+x^3)+x^3 \times (-2 x^4+x^3)\\\\=3 \times -2\times x^4 \times x^4 +3 \times 1\times x^4 \times x^3 + x^3 \times x^4 \times -2+x^3 \times x^3\\\\=-6 x^8+3 x^7-2 x^7+x^6\\\\=-6 x^8+ x^7+x^6$

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

$=(x^2+2 x)(x+1)\\\\= x^2\times(x+1)+2 x\times (x+1)\\\\= x^3+x^2+2 x^2+2 x\\\\= x^3+3x^2+2 x=RHS$  

→→→True

Option B: is not correctly closed under Multiplication.