Упр.739 ГДЗ Мерзляк Полонский 7 класс (Алгебра)

(а2 + b2 + с2) (m2 + n2 + k2) — (am + bn + ck)2 = = (an — bm)2 + (ak — cm)2 + (bk — cn)2.
Разложите на множители:
1)х4-5х2+4;
2) х4 + х2 + 1;
3) 4х4 -12х2 + 1;
4) х5 + х + 1;
5) х4 + 4;
6)х8+х4-2.

1) Докажем тождество.

$$ (a^2+b^2+c^2)(m^2+n^2+k^2)-(am+bn+ck)^2 $$
$$ =a^2m^2+a^2n^2+a^2k^2+b^2m^2+b^2n^2+b^2k^2+c^2m^2+c^2n^2+c^2k^2 $$
$$ -a^2m^2-b^2n^2-c^2k^2-2abmn-2ackm-2bcnk $$
$$ =a^2n^2+a^2k^2+b^2m^2+b^2k^2+c^2m^2+c^2n^2-2abmn-2ackm-2bcnk $$
$$ =(an-bm)^2+(ak-cm)^2+(bk-cn)^2. $$

2) Разложим на множители.

1. $$ x^4-5x^2+4=x^4-4x^2+4-x^2=(x^2-2)^2-x^2 $$
   $$ =(x^2-2-x)(x^2-2+x)=(x^2-x-2)(x^2+x-2) $$
   $$ =(x-2)(x+1)(x-1)(x+2). $$

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

3. $$ 4x^4-12x^2+1=4x^4+4x^2+1-16x^2=(2x^2+1)^2-(4x)^2 $$
   $$ =(2x^2+1-4x)(2x^2+1+4x). $$

4. $$ x^5+x+1=x^5-x^2+x^2+x+1=x^2(x^3-1)+(x^2+x+1) $$
   $$ =x^2(x-1)(x^2+x+1)+(x^2+x+1) $$
   $$ =(x^2+x+1)\bigl(x^2(x-1)+1\bigr) =(x^2+x+1)(x^3-x^2+1). $$

5. $$ x^4+4=x^4+4x^2+4-4x^2=(x^2+2)^2-(2x)^2 $$
   $$ =(x^2+2-2x)(x^2+2+2x). $$

6. $$ x^8+x^4-2=x^8-2x^4+1+3x^4-3=(x^4-1)^2+3(x^4-1) $$
   $$ =(x^4-1)(x^4+2)=(x^2-1)(x^2+1)(x^4+2) $$
   $$ =(x-1)(x+1)(x^2+1)(x^4+2). $$

Ответ

$$ (a^2+b^2+c^2)(m^2+n^2+k^2)-(am+bn+ck)^2=(an-bm)^2+(ak-cm)^2+(bk-cn)^2 $$

1) $$(x-2)(x+1)(x-1)(x+2)$$

2) $$(x^2+1-x)(x^2+1+x)$$

3) $$(2x^2+1-4x)(2x^2+1+4x)$$

4) $$(x^2+x+1)(x^3-x^2+1)$$

5) $$(x^2+2-2x)(x^2+2+2x)$$

6) $$(x-1)(x+1)(x^2+1)(x^4+2)$$