Question

помогите пожалуйста!!
даю 35 баллов​

Answer 1

$5)\ \ (x-p)(x^3+x^2-1)=x^4+x^3-x-px^3-px^2+p=\\\\=x^4+(-1-p)\, x^3+(-p)\, x^2-x+p\\\\a)\ \ -1-p=-1\ \ ,\ \ p=-1+1\ \ ,\ \ \underline {\ p=0\ }\\\\b)\ \ -p=0\ \ ,\ \ \underline {\ p=0\ }$

$6)\ \ (a^2+b^2+ab+1)(a^2+b^2-ab+1)-(a^2+b^2+1)^2=\\\\\Big[\ t=a^2+b^2+1\ \Big]=(t+ab)(t-ab)-t^2=t^2-(ab)^2-t^2=-(ab)^2=-a^2b^2$

$7)\ \ (a+b)(a^3-a^2b+ab^2-b^3)=\Big(a+b\Big)\Big((a^3-b^3)-ab(a-b)\Big)=\\\\=\Big(a+b\Big)\Big(a-b\Big)\Big((a^2+ab+b^2)-ab\Big)=\Big(a+b\Big)\Big(a-b\Big)\Big(a^2+b^2\Big)\\\\\\\\(a-b)(a^2+a^2b+ab^2+b^3)=\Big(a-b\Big)\Big((a^3+b^3)+ab(a+b)\Big)=$

$=\Big(a-b\Big)\Big(a+b\Big)\Big((a^2-ab+b^2)+ab\Big)=\Big(a-b\Big)\Big(a+b\Big)\Big(a^2+b^2\Big)$

$(a+b)(a^3-a^2b+ab^2-b^3)\equiv (a-b)(a^3+a^2b+ab^2+b^3)\\\\\\\star \ \ \ a^3+b^3=(a+b)(a^2-ab+b^2)\ \ \star \\\\\star \ \ a^3-b^3=(a+b)(a^2+ab+b^2)\ \ \star$