Question

Запишите многочлен в стандартном виде:
a) $(x^{2}-3x+2)^{2}-(x^{2}-x)^{2}$
b) $(x+1)(x^{7}-x^{6}+...-x^{2}+x-1)$
c) $(2-x)^{3}+(x-1)^{3}$
d) $(x^{5}-x^{4}+x^{3}-x^{2}+x-1)(x^{5}+x^{4}+x^{3}+x^{2}+x+1)$

Answer 1

a) $(x^{2}-3x+2)^{2}-(x^{2}-x)^{2} =$
$= (x^{2}-3x+2 - x^{2}+x )(x^{2}-3x+2 + x^{2}-x) =$
$=(-2x+2)(2x^{2}-4x+2)=$
$= -4x^{3} + 12x^{2}-12x+4$
b) $(x+1)(x^{7}-x^{6}+...-x^{2}+x-1) =$
$= x^{8} - x^{7}+...-x^{3}+x^{2}-x+x^{7}-x^{6}+...-x^{2}+x-1=x^{8} - 1$
c) $(2-x)^{3}+(x-1)^{3}=8 - 12x + 6x^{2} - x^{3} + x^{3} - 3x^{2} + 3x - 1=$
$=3x^{2} - 9x + 7$
d) $(x^{5}-x^{4}+x^{3}-x^{2}+x-1)(x^{5}+x^{4}+x^{3}+x^{2}+x+1)=$$=(x^{4}(x-1)+x^{2}(x-1)+(x-1))(x^{4}(x+1)+x^{2}(x+1)+(x+1))=$
$=(x-1)(x^{4}+x^{2}+1)(x + 1)(x^{4}+x^{2}+1)=(x^2 - 1)(x^{4}+x^{2}+1)^{2}=$
$=(x^8+2x^6+3x^4+2x^2+1)(x^2-1)=$
$=x^{10}+2x^8+3x^6+2x^4+x^2-x^8-2x^6-3x^4-2x^2-1=$$=x^{10}+x^8+x^6-x^4-x^2-1$