Упр.29 Задачи повышенной сложности ГДЗ Колмогоров 10-11 класс (Алгебра)
а) x^4+4; б) x^4+x^2+1; в) x^5+x+1;
г) (x^2+y^2)^3+(z^2-x^2)^3-(y^2+z^2)^3;
д) (x+y+z)^3-x^3-y^3-z^3; е) x^3+y^3+z^3-3xyz.
а) $$x^4+4=(x^2+2)^2-4x^2=(x^2-2x+2)(x^2+2x+2).$$
б) $$x^4+x^2+1=(x^2+1)^2-x^2=(x^2-x+1)(x^2+x+1).$$
в) $$x^5+x+1=x^5-x^2+x^2+x+1=x^2(x^3-1)+(x^2+x+1)$$
$$=x^2(x-1)(x^2+x+1)+(x^2+x+1)=(x^2+x+1)(x^3-x^2+1).$$
г) $$\begin{aligned} &(x^2+y^2)^3+(z^2-x^2)^3-(y^2+z^2)^3 \\ &=(y^2+z^2)\bigl((x^2+y^2)^2-(x^2+y^2)(z^2-x^2)+(z^2-x^2)^2\bigr)-(y^2+z^2)^3 \\ &=(y^2+z^2)\bigl((x^2+y^2)^2-(x^2+y^2)(z^2-x^2)-(x^2+y^2)(2z^2+y^2-x^2)\bigr) \\ &=(y^2+z^2)(x^2+y^2)(3x^2-3z^2) \\ &=3(y^2+z^2)(x^2+y^2)(x-z)(x+z). \end{aligned}$$
д) $$\begin{aligned} &(x+y+z)^3-x^3-y^3-z^3 \\ &=(y+z)\bigl((x+y+z)^2+(x+y+z)x+x^2\bigr)-(y+z)(y^2-yz+z^2) \\ &=(y+z)\bigl((x+y+z)^2+(x+y+z)x+(x-y)(x+y)+z(y-z)\bigr) \\ &=(y+z)(3x^2+3xy+3xz+3yz) \\ &=3(y+z)(x+z)(x+y). \end{aligned}$$
е) $$\begin{aligned} &x^3+y^3+z^3-3xyz \\ &=(x+y)^3-3xy(x+y)+z^3-3xyz \\ &=(x+y+z)\bigl((x+y)^2-(x+y)z+z^2\bigr)-3xy(x+y+z) \\ &=(x+y+z)(x^2+2xy+y^2-xz-yz+z^2-3xy) \\ &=(x+y+z)(x^2+y^2+z^2-xy-yz-xz). \end{aligned}$$
Ответ
а) $$\left(x^2-2x+2\right)\left(x^2+2x+2\right);$$ б) $$\left(x^2-x+1\right)\left(x^2+x+1\right);$$ в) $$\left(x^2+x+1\right)\left(x^3-x^2+1\right);$$ г) $$3(y^2+z^2)(x^2+y^2)(x-z)(x+z);$$ д) $$3(x+y)(x+z)(y+z);$$ е) $$\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right).$$
