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1) B=(x-18)(x-7)(x+35)(x+90)-67x^2
2) a(b^2-c^2)-b(a^2-c^2)+c(a^2-b^2)
3) a^3(b-c)+b^3(c-a)+c^3(a-b)
4) a(b+c)(b^2-c^2)+b(a+c)(c^2-b^2)+c(a+b)(a^2-b^2)
5)a^2(b-2c)+b^2(c-a)+2c^2(a-b)+abc

 

[yennhi1312]

1) B=(x-18)(x-7)(x+35)(x+90)-67x^2
2) a(b^2-c^2)-b(a^2-c^2)+c(a^2-b^2)
3) a^3(b-c)+b^3(c-a)+c^3(a-b)
4) a(b+c)(b^2-c^2)+b(a+c)(c^2-b^2)+c(a+b)(a^2-b^2)
5)a^2(b-2c)+b^2(c-a)+2c^2(a-b)+abc

1) $B=(x-18)(x-7)(x+35)(x+90)-67x^2$
$=[(x-18)(x+35)][(x-7)(x+90)]-67x^2$
$=(x^2+17x-630)(x^2+83x-630)-67x^2$
Đặt $x^2+17x-630=y\Rightarrow$ bt $=y(y+66x)-67x^2=(y-x)(y+67x)$.
Thay $y=x^2+17x-630$ ta được bt $=(x^2+17x-630-x)(x^2+17x-630+67x)\\=(x^2+16x-630)(x^2+84x-630)$
2) $a(b^2-c^2)-b(a^2-c^2)+c(a^2-b^2)$
$=ab^2-ac^2-a^2b+bc^2+c(a+b)(a-b)$
$=-(a^2b-ab^2)-(ac^2-bc^2)+(ac+bc)(a-b)$
$=-ab(a-b)-c^2(a-b)+(ac+bc)(a-b)$
$=(a-b)(bc-c^2-ab+ac)$
$=(a-b)[c(b-c)-a(b-c)]$
$=(a-b)(b-c)(c-a)$
3) $a^3(b-c)+b^3(c-a)+c^3(a-b)$
$=a^3[(a-c)-(a-b)]-b^3(a-c)+c^3(a-b)$
$=a^3(a-c)-a^3(a-b)-b^3(a-c)+c^3(a-b)$
$=(a-c)(a^3-b^3)-(a-b)(a^3-c^3)$
$=(a-c)(a-b)(a^2+ab+b^2)-(a-b)(a-c)(a^2+ac+c^2)$
$=(a-b)(a-c)(b^2-c^2+ab-ac)$
$=(a-b)(a-c)[(b-c)(b+c)+a(b-c)]$
$=(a-b)(b-c)(a-c)(a+b+c)$

 

[Blue Plus]

1) B=(x-18)(x-7)(x+35)(x+90)-67x^2
2) a(b^2-c^2)-b(a^2-c^2)+c(a^2-b^2)
3) a^3(b-c)+b^3(c-a)+c^3(a-b)
4) a(b+c)(b^2-c^2)+b(a+c)(c^2-b^2)+c(a+b)(a^2-b^2)
5)a^2(b-2c)+b^2(c-a)+2c^2(a-b)+abc

5.
$ a^2(b - 2c) + b^2(c - a) + 2c^2(a - b) + abc \\ = a^2b - 2a^2c + b^2c - ab^2 + 2ac^2 - 2bc^2 + abc \\ = (2ac^2 - abc) + (-2bc^2 + b^2c) + (-2a^2c + a^2b) + (2abc - ab^2) \\ = ac(2c - b) - bc(2c - b) - a^2(2c - b) + ab(2c - b) \\ = (2c - b)(ac - bc - a^2 + ab) \\ = (2c - b)[c(a - b) - a(a - b)] \\ = (2c - b)(a - b)(c - a) $

 

[Mai Hải Đăng]

4/[tex]a(b+c)(b^{2}-c^{2})+b(a+c)(c^{2}-b^{2})+c(a+b)(a^{2}-b^{2})[/tex]
[tex]=ab^{3}-abc^{2}+ab^{2}c - ac^{3}+abc^{2}-ab^{3}+bc^{3}-b^{3}c+a^{3}c-ab^{2}c+a^{2}bc-b^{3}c[/tex]
[tex]= -(ac^3-bc^3)+(a^2bc-b^3c)+(a^3c-b^3c)[/tex]
[tex]=-c^{3}(a-b)+bc(a^{2}-b^{2})+c(a^{3}-b^{3})[/tex]
[tex]=-c^3(a-b)+bc(a-b)(a+b)+c(a-b)(a^2+ab+b^{2})[/tex]
[tex]=c(a-b)(-c^{2}+ab+b^2+a^2+ab+b^2)[/tex]

 

[Blue Plus]

4/[tex]a(b+c)(b^{2}-c^{2})+b(a+c)(c^{2}-b^{2})+c(a+b)(a^{2}-b^{2})[/tex]
[tex]=[COLOR=#ff0000]ab^{3}[/COLOR][COLOR=#0000ff]-abc^{2}[/COLOR][COLOR=#ff00ff]+ab^{2}c[/COLOR] - ac^{3}[COLOR=#0000ff]+abc^{2}[/COLOR][COLOR=#ff0000]-ab^{3}[/COLOR]+bc^{3}-b^{3}c+a^{3}c[COLOR=#ff00ff]-ab^{2}c[/COLOR]+a^{2}bc-b^{3}c[/tex]
[tex]= - ([B]ac^{3}-bc^{3[/B])+([B]a^{2}bc-b^{3}c[/B])+([B]a^{3}c[B][B]-b^{3}c[/B][/B][/B])[/tex]
[tex]=-c^{3}(a-b)+bc(a^{2}-b^{2})+c(a^{3}-b^{3})[/tex]
[tex]=-c^3(a-b)+bc(a-b)(a+b)+c(a-b)(a^2+ab+b^{2})[/tex]
[tex]=c(a-b)(-c^{2}+ab+b^2+a^2+ab+b^2)[/tex]

Bn thu gọn thành $ c(a - b) (-c^2 + 2ab + 2b^2 + a^2 $
Lỗi latex phần giữa