Упр.343 ГДЗ Колягин Ткачёва 10 класс (Алгебра)

1) (y — z)(y + z)2 + (z + x)(z + x)2 + (х- у)(х + y)2;
2) x6- y6 + (x4 + x2y2 + y4);
3) х8 + х4y4 + y8.

1. $$\begin{aligned} (y-z)(y+z)^2+(z+x)(z+x)^2+(x-y)(x+y)^2 &= (y-z)(y^2+2yz+z^2)+(z+x)(z^2+2zx+x^2)\\ &\quad +(x-y)(x^2+2xy+y^2)\\ &= y^3+y^2z-yz^2-z^3+z^3+z^2x+zx^2+x^3\\ &\quad -y^3-y^2x+xy^2+x^3\\ &= y^2z-yz^2+z^2x-zx^2+xy^2-x^2y\\ &= (y-z)(yz-zx+xy)-x(y-z)(x+y)\\ &= (y-z)\bigl(yz-zx+xy-x(x+y)\bigr)\\ &= (y-z)(yz-zx-zx)\\ &= (y-z)(z-x)(y-x). \end{aligned}$$

2. $$\begin{aligned} x^6-y^6+(x^4+x^2y^2+y^4) &= (x^2-y^2)(x^4+x^2y^2+y^4)+(x^4+x^2y^2+y^4)\\ &= (x^4+x^2y^2+y^4)(x^2-y^2+1)\\ &= (x^2+y^2-xy)(x^2+y^2+xy)(x^2-y^2+1). \end{aligned}$$

3. $$\begin{aligned} x^8+x^4y^4+y^8 &= (x^4+y^4)^2-(x^2y^2)^2\\ &= (x^4+y^4-x^2y^2)(x^4+y^4+x^2y^2). \end{aligned}$$

Ответ

$$1)\ (y-x)(y-z)(z-x);$$
$$2)\ (x^2-y^2+1)(x^2+y^2-xy)(x^2+y^2+xy);$$
$$3)\ (x^4+y^4-x^2y^2)(x^4+y^4+x^2y^2).$$