\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\frac{1}{a+b+c}=\left(\frac{1}{a}+\frac{1}{b}\right)+\left(\frac{1}{c}-\frac{1}{a+b+c}\right)\)

\(=\frac{a+b}{ab}+\frac{a+b}{c\left(a+b+c\right)}=\left(a+b\right)\left(\frac{1}{ab}+\frac{1}{c\left(a+b+c\right)}\right)=\frac{\left(a+b\right)\left(ab+ac+bc+c^2\right)}{abc\left(a+b+c\right)}\)

\(=\frac{\left(a+b\right)\left(a+c\right)\left(b+c\right)}{abc\left(a+b+c\right)}\)

\(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}-\frac{3}{abc}=\left(\frac{1}{a}+\frac{1}{b}\right)^3+\left(\frac{1}{c}\right)^3-3.\frac{1}{a}.\frac{1}{b}\left(\frac{1}{a}+\frac{1}{b}\right)-\frac{3}{abc}\)

\(=\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\left[\left(\frac{1}{a}+\frac{1}{b}\right)^2-\left(\frac{1}{a}+\frac{1}{b}\right).\frac{1}{c}+\frac{1}{c^2}\right]-3.\frac{1}{a}.\frac{1}{b}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)

\(=\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{2}{ab}-\frac{1}{ac}-\frac{1}{bc}+\frac{1}{c^2}\right)-\frac{3}{ab}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)

\(=\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}-\frac{1}{ab}-\frac{1}{ac}-\frac{1}{bc}\right)\)

a) \(x^3+y^3+z^3-3xyz\)

\(=x^3+3x^2y+3xy^2+y^3+z^3-3x^2y-3xy^3-3xyz\)

\(=\left(x+y\right)^3+z^3-3xy\left(x+y+z\right)\)

\(=\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right)z+z^2\right]-3xy\left(x+y+z\right)\)

\(=\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2\right)-3xy\left(x+y+z\right)\)

\(=\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2-3xy\right)\)

\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-xz-yz\right)\)

câu b đâu

Bài 1 :

a) \(x^8+x+1\)

\(=x^8-x^2+\left(x^2+x+1\right)\)

\(=x^2\left(x^6-1\right)+\left(x^2+x+1\right)\)

\(=x^2\left(x^3+1\right)\left(x^3-1\right)+\left(x^2+x+1\right)\)

\(=\left(x^5+x^2\right)\left(x^3-1\right)+\left(x^2+x+1\right)\)

\(=\left(x^5+x^2\right)\left(x-1\right)\left(x^2+x+1\right)+\left(x^2+x+1\right)\)

\(=\left(x^6-x^5+x^3-x^2\right)\left(x^2+x+1\right)+\left(x^2+x+1\right)\)

\(=\left(x^6-x^5+x^4-x^2+1\right)\left(x^2+x+1\right)\)

b) \(64x^4+y^4\)

\(=\left(8x^2\right)^2+\left(y^2\right)^2+2.8x^2.y^2-16x^2y^2\)

\(=\left(8x^2+y^2\right)^2-\left(4xy\right)^2\)

\(=\left(8x^2+y^2-4xy\right)\left(8x^2+y^2+4xy\right)\)

Bạn thử giải câu này xem

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\(x\left(x+2\right)\left(x^2+2x+2\right)+1\)

\(=\left(x^2+2x\right)\left(x^2+2x+2\right)+1\)

Đặt: \(x^2+2x=t\)

khi đó: \(\left(x^2+2x\right)\left(x^2+2x+2\right)+1=t\left(t+2\right)+1=\left(t+1\right)^2\)

\(=\left(x^2+2x+1\right)^2=\left(x+1\right)^4\)

b) Xét: \(\left(n+1\right)^2-n^2=\left(n+1+n\right)\left(n+1-n\right)=2n+1\)

Khi đó:

\(A=\frac{3}{\left(1.2\right)^2}+\frac{5}{\left(2.3\right)^2}+\frac{7}{\left(3.4\right)^2}+...+\frac{2n+1}{\left[n\left(n+1\right)\right]^2}\)

\(A=\frac{2^2-1^2}{1^2.2^2}+\frac{3^2-2^2}{2^2.3^2}+\frac{4^2-3^2}{3^2.4^2}+...+\frac{\left(n+1\right)^2-n^2}{n^2.\left(n+1\right)^2}\)

\(A=1-\frac{1}{2^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{3^2}-\frac{1}{4^2}+...+\frac{1}{n^2}-\frac{1}{\left(n+1\right)^2}\)

\(A=1-\frac{1}{\left(n+1\right)^2}\)

2.

pt <=> (x/2000 - 1) + (x+1/2001 - 1) + (x+2/2002 - 1) + (x+3/2003 - 1) + (x+4/2004 - 1 ) = 0

<=> x-2000/2000 + x-2000/2001 + x-2000/2002 + x-2000/2003 + x-2000/2004 = 0

<=> (x-2000).(1/2000 + 1/2001 + 1/2002 + 1/2003 + 1/2004) = 0

<=> x-2000=0 ( vì 1/2000 + 1/2001 + 1/2002 + 1/2003 + 1/2004 > 0 )

<=> x=2000

Tk mk nha

1.

a, = (2x-1)^2-2.(2x-1)+1-4

    = (2x-1-1)^2-4

    = (2x-2)^2-4

    = (2x-2-2).(2x-2+2)

    = 2x.(2x-4)

b, = [x.(x+3)].[(x+1).(x+2)]

    = (x^2+3x).(x^2+3x+1)-8

    = (x^2+3x+1)^2-1-8

    = (x^2+3x+1)^2-9

    = (x^2+3x+1-3).(x^2+3x+1+3)

    = (x^2+3x-2).(x^2+3x+4)

    = ((x+1).(x+3).(x^2+3x-2)

Tk mk nha

bạn có thể giải giùm  mk bài toán hình nx k ạ

Mấy bài này mình đã làm rồi. 

\(\frac{2}{xy}:\left(\frac{1}{x}-\frac{1}{y}\right)^2-\frac{x^2+y^2}{\left(x-y\right)^2}\)

\(=\frac{2}{xy}:\left(\frac{y-x}{xy}\right)^2-\frac{x^2+y^2}{\left(x-y\right)^2}\)

\(=\frac{2}{xy}:\frac{\left(x-y\right)^2}{x^2y^2}-\frac{x^2+y^2}{\left(x-y\right)^2}\)

\(=\frac{2x^2y^2}{xy\left(x-y\right)^2}-\frac{x^2+y^2}{\left(x-y\right)^2}\)

\(=\frac{2xy}{\left(x-y\right)^2}-\frac{x^2+y^2}{\left(x-y\right)^2}=\frac{-x^2+2xy-y^2}{\left(x-y\right)^2}\)

\(=-\frac{\left(x-y\right)^2}{\left(x-y\right)^2}=-1\)

a.(x+1)(x+2)(x+3)(x+4)-24=[(x+1)(x+4)][(x+2)(x+3)]-24=(\(x^2+5x+4\))(\(x^2+5x+6\))-24  (1)

đặt \(x^2+5x+5=a\)ta có (1)=(a-1)(a+1)-24=\(a^2-25=\left(a-5\right)\left(a+5\right)\)

thay a=\(x^2+5x+5\)vào (1) ta có (1)=(\(x^2+5x\)+5-5)(\(x^2+5x\)+5+5)=x(x+5)(\(x^2\)+5x+10)

b.ta có :\(\frac{a}{3}+\frac{a^2}{2}+\frac{a^3}{6}=\frac{2a+3a^2+a^3}{6}=\frac{a\left(a^2+3a+2\right)}{6}\)=\(\frac{a\left(a^2+2a+a+2\right)}{6}=\frac{a\left(a+1\right)\left(a+2\right)}{6}\).ta lại có a(a+1)(a+2) là tích 3 số nguyên liên tiếp luôn chia hết cho 6 suy ta điều cần cm