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PTĐTTNT?
1.Đặt \(a^2+a=t\)
\(\Rightarrow\left(a^2+a\right)\left(a^2+a+1\right)-2\)
\(=t\left(t+1\right)-2\)
\(=t^2+t-2\)
\(=t^2+2t-\left(t+2\right)\)
\(=t\left(t+2\right)-\left(t+2\right)\)
\(=\left(t+2\right)\left(t-1\right)\)
Sửa đề:
\(x^4+2011x^2+2010x+2011\)
\(=\left(x^4-x\right)+2011x^2+2011x+2011\)
\(=x\left(x^3-1\right)+2011\left(x^2+x+1\right)\)
\(=x\left(x-1\right)\left(x^2+x+1\right)+2011\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^2-x+2011\right)\)
3. \(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-120\)
\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)-120\)
Đặt \(x^2+5x+4=t\)
\(\Rightarrow\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-120\)
\(=t\left(t+2\right)-120\)
\(=t^2+2t+1-121\)
\(=\left(t+1\right)^2-11^2\)
\(=\left(t+1-11\right)\left(t+1+11\right)\)
\(=\left(t-10\right)\left(t+12\right)\)
\(=\left(x^2+5x-6\right)\left(x^2+5x+16\right)\)
\(=\left[\left(x^2-x\right)+\left(6x-6\right)\right]\left(x^2+5x+16\right)\)
\(=\left[x.\left(x-1\right)+6\left(x-1\right)\right]\left(x^2+5x+16\right)\)
\(=\left(x-1\right)\left(x+6\right)\left(x^2+5x+16\right)\)
4. \(\left(x^2+x+4\right)^2+8x\left(x^2+x+1\right)+15x^2\)
\(=\left(x^2+x+4\right)^2+2.\left(x^2+x+1\right).4x+\left(4x\right)^2-x^2\)
\(=\left(x^2+x+4+4x\right)^2-x^2\)
\(=\left(x^2+4+5x-x\right)\left(x^2+5x+x+4\right)\)
\(=\left(x^2+4x+4\right)\left(x^2+6x+4\right)\)
\(=\left(x+2\right)^2\left[\left(x^2+2.x.3+3^2\right)-\left(\sqrt{5}\right)^2\right]\)
\(=\left(x+2\right)^2\left[\left(x+3\right)^2-\left(\sqrt{5}\right)^2\right]\)
\(=\left(x+2\right)^2\left(x+3-\sqrt{5}\right)\left(x+3+\sqrt{5}\right)\)
A=(\(\frac{1}{X^3}\)+x3)+(\(\frac{1}{y^3}\)+y3)+(\(\frac{1}{z^3}\)+z3)+3
Áp dung bđt AM-GM(Cosi) cho hai số dương lần lượt ta đc
A>=6khi x=1,y1,z=1
1. ( 2x + y )( 4x2 - 2xy + y2 ) - 8x3 - y3 - 16
= [ ( 2x )3 + y3 ] - 8x3 - y3 - 16
= 8x3 + y3 - 8x3 - y3 - 16
= -16 ( đpcm )
2. ( 3x + 2y )2 + ( 3x + 2y )2 - 18x2 - 8y2 + 3
= 2( 3x + 2y )2 - 18x2 - 8y2 + 3
= 2( 9x2 + 12xy + 4y2 ) - 18x2 - 8y2 + 3
= 18x2 + 24xy + 8y2 - 18x2 - 8y2 + 3
= 24xy + 3 ( có phụ thuộc vào biến )
3. ( -x - 3 )3 + ( x + 9 )( x2 + 27 ) + 19
= -x3 - 9x2 - 27x - 27 + x3 + 9x2 + 27x + 243 + 19
= -27 + 243 + 19 = 235 ( đpcm )
4. ( x - 2 )3 - x( x + 1 )( x - 1 ) + 13( x - 4 )
= x3 - 6x2 + 12x - 8 - x( x2 - 1 ) + 13x - 52
= x3 - 6x2 + 12x - 8 - x3 + x + 13x - 52
= -6x2 + 26x - 60 ( có phụ thuộc vào biến )