Упр.317 ГДЗ Никольский Потапов 7 класс (Алгебра)

б) (х2 — 1)(х4 + х2 + 1) — (х2 — 1) (х2 — 1) (х2 — 1);
в) (m+1/2)(m2-1/2*m+1/4) — (1/2*m3-1/2*m2);
г) (1/2*a-2b)(1/3*a4+ab+4b2)-(1/8*a3-8b3);
д) 15x3y2 — (5ху — 2)(3x2y + x);
е) 1/2(a + b + c)(a + b — c) — ab;
ж) (a + 2b) (a + c) — (a — 2b) (a — c).

1. а)

   $$ (a^2+1)(a^2+1)+(a-1)(a^2+1)-a^2 $$
   $$ =a^4+a^2+a^2+1+a^3+a-a^2-1-a^2 $$
   $$ =a^4+a^3+a. $$

2. б)

   $$ (x^2-1)(x^4+x^2+1)-(x^2-1)(x^2-1)(x^2-1) $$
   $$ =x^6+x^4+x^2-x^4-x^2-1-(x^4-2x^2+1)(x^2-1) $$
   $$ =x^6-1-(x^6-3x^4+3x^2-1) $$
   $$ =x^6-1-x^6+3x^4-3x^2+1 $$
   $$ =3x^4-3x^2. $$

3. в)

   $$ \left(m+\frac12\right)\left(m^2-\frac12m+\frac14\right)-\left(\frac12m^3-\frac12m^2\right) $$
   $$ =m^3-\frac12m^2+\frac14m+\frac12m^2-\frac14m+\frac18-\frac12m^3+\frac12m^2 $$
   $$ =\frac12m^3+\frac12m^2+\frac18. $$

4. г)

   $$ \left(\frac12a-2b\right)\left(\frac14a^2+ab+4b^2\right)-\left(\frac18a^3-8b^3\right) $$
   $$ =\frac18a^3+\frac12a^2b+2ab^2-\frac12a^2b-2ab^2-8b^3-\frac18a^3+8b^3 $$
   $$ =0. $$

5. д)

   $$ 15x^3y^2-(5xy-2)(3x^2y+x) $$
   $$ =15x^3y^2-(15x^3y^2+5x^2y-6x^2y-2x) $$
   $$ =15x^3y^2-15x^3y^2+x^2y+2x $$
   $$ =x^2y+2x. $$

6. е)

   $$ \frac12(a+b+c)(a+b-c)-ab $$
   $$ =\frac12\bigl((a+b)^2-c^2\bigr)-ab $$
   $$ =\frac12(a^2+2ab+b^2-c^2)-ab $$
   $$ =\frac12a^2+\frac12b^2-\frac12c^2. $$

7. ж)

   $$ (a+2b)(a+c)-(a-2b)(a-c) $$
   $$ =a^2+ac+2ab+2bc-\bigl(a^2-ac-2ab+2bc\bigr) $$
   $$ =2ac+4ab. $$

Ответ

а) $a^4+a^3+a$; б) $3x^4-3x^2$; в) $\frac12m^3+\frac12m^2+\frac18$; г) $0$; д) $x^2y+2x$; е) $\frac12a^2+\frac12b^2-\frac12c^2$; ж) $2ac+4ab$.