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Bài 1:
\(a,\left(x^2-1\right)^3-\left(x^4+x^2+1\right)\left(x^2-1\right)\)
\(=x^6-3x^4+3x^2-1-x^6+1\)
\(=-3x^2\left(x^2-1\right)\)
\(b,\left(x^4-3x^2+9\right)\left(x^2+3\right)-\left(3+x^2\right)^3\)
\(=x^6+27-27-27x^2-9x^4-x^6\)
\(=-9x^2\left(3-x^2\right)\)
Bài 5:
\(A=x^2-2x+1\)
\(=\left(x^2-2x+1\right)-2\)
\(=\left(x-1\right)^2-2\)
Với mọi giá trị của x ta có:
\(\left(x-1\right)^2\ge0\Rightarrow\left(x-1\right)^2-2\ge-2\)
Vậy Min A = -2
Để A = -2 thì \(x-1=0\Rightarrow x=1\)
b, \(B=4x^2+4x+5\)
\(=\left(4x^2+4x+1\right)+4\)
\(=\left(2x+1\right)^2+4\)
Với mọi giá trị của x ta có:
\(\left(2x+1\right)^2\ge0\Rightarrow\left(2x+1\right)^2+4\ge4\)
Vậy Min B = 4
Để B = 4 thì \(2x+1=0\Rightarrow2x=-1\Rightarrow x=-\dfrac{1}{2}\)
c, \(C=2x-x^2-4\)
\(=-\left(x^2-2x+1\right)-3\)
\(=-\left(x-1\right)^2-3\)
Với mọi giá trị của x ta có:
\(\left(x-1\right)^2\ge0\Rightarrow-\left(x-1\right)^2\le0\Rightarrow-\left(x-1\right)^2-3\le-3\)Vậy Max C = -3
để C = -3 thì \(x-1=0\Rightarrow x=1\)
tìm a,b,c biết rằng P(x) = x^3 + ax^2 +bx + c chia hết cho x-2 và chia x^2 -1 dư 2x
GIÚP MÌNH NHA!...
\(A=x\left(x+2\right)+y\left(y-2\right)-2xy+37\)
\(=x^2+2x+y^2-2y-2xy+37\)
\(=\left(x^2-2xy+y^2\right)+2\left(x-y\right)+37\)
\(=\left(x-y\right)^2+2\left(x-y\right)+37\)
Thay x - y = 7
\(\Rightarrow A=49+14+37=100\)
Vậy A = 100 khi x - y = 7
\(x^4-x^3-2x-4\)
\(=x^4-x^3-2x^2+2x^2-2x-4\)
\(=x^2\left(x^2-x-2\right)+2\left(x^2-x-2\right)\)
\(=\left(x^2-x-2\right)\left(x^2+2\right)\)
\(=\left(x^2+x-2x-2\right)\left(x^2+2\right)\)
\(=\left[x\left(x+1\right)-2\left(x+1\right)\right]\left(x^2+2\right)\)
\(=\left(x-2\right)\left(x+1\right)\left(x^2+2\right)\)
a,\(\frac{3}{2x^2+2x}+\frac{2x-1}{x^2-1}-\frac{2}{x}\)
\(=\frac{3}{2x\left(x+1\right)}+\frac{2x-1}{\left(x-1\right)\left(x+1\right)}-\frac{2}{x}\)
\(=\frac{3\left(x-1\right)}{2x\left(x+1\right)\left(x-1\right)}+\frac{\left(2x-1\right).2x}{2x\left(x-1\right)\left(x+1\right)}-\frac{2.2\left(x+1\right)\left(x-1\right)}{2x\left(x+1\right)\left(x-1\right)}\)
\(=\frac{3x-3}{2x\left(x+1\right)\left(x-1\right)}+\frac{4x^2-2x}{2x\left(x-1\right)\left(x+1\right)}-\frac{4x^2-4}{2x\left(x+1\right)\left(x-1\right)}\)
\(=\frac{3x-3+4x^2-2x-4x^2+4}{2x\left(x+1\right)\left(x-1\right)}\)
\(=\frac{x+1}{2x\left(x+1\right)\left(x-1\right)}=\frac{1}{2x\left(x-1\right)}\)
\(b,\frac{3x}{5x+5y}-\frac{x}{10x-10y}\)
\(=\frac{3x}{5\left(x+y\right)}-\frac{x}{10\left(x-y\right)}\)
\(=\frac{3x.2\left(x-y\right)}{10\left(x+y\right).\left(x-y\right)}-\frac{x.\left(x+y\right)}{10\left(x-y\right).\left(x+y\right)}\)
\(=\frac{6x^2-6xy}{10\left(x+y\right)\left(x-y\right)}-\frac{x^2+xy}{10\left(x-y\right)\left(x+y\right)}\)
\(=\frac{6x^2-6xy-x^2+xy}{10\left(x+y\right)\left(x-y\right)}\)
\(=\frac{5x^2-5xy}{10\left(x+y\right)\left(x+y\right)}\)
\(=\frac{5x\left(x-y\right)}{10\left(x-y\right)\left(x+y\right)}=\frac{x}{2\left(x+y\right)}\)
\(a,2x^2+8x+5\)
\(=\left(\sqrt{2}x\right)^2+2.\sqrt{2}x.\dfrac{8}{2\sqrt{2}}+\left(\dfrac{8}{2\sqrt{2}}\right)^2-\left(\dfrac{8}{2\sqrt{2}}\right)^2+5\)
\(=\left[\left(\sqrt{2}x\right)^2+2.\sqrt{2}x.\dfrac{8}{2\sqrt{2}}+\left(\dfrac{8}{2\sqrt{2}}\right)^2\right]-\left(\dfrac{8}{2\sqrt{2}}\right)^2+5\)
\(=\left(\sqrt{2}x+\dfrac{8}{2\sqrt{2}}\right)^2-3\)
Ta có :
\(\left(\sqrt{2}x+\dfrac{8}{2\sqrt{2}}\right)^2\ge0\forall x\)
\(\Rightarrow\left(\sqrt{2}x+\dfrac{8}{2\sqrt{2}}\right)^2-3\ge-3>0\)
Dấu = xảy ra khi \(\sqrt{2}x+\dfrac{8}{2\sqrt{2}}=0\Rightarrow x=-2\)
Các câu còn lại dễ rồi mk ko lm nx nha bn ,bn ko bt lm cỗ nào thì hỏi mk
\(z^4-4z^3+z^2+4z^2-4z+1\)
\(=z^4-4z^3+z^2+4z^2-4z+1\)
\(=\left(z^4-4z^3+z^2\right)+\left(4z^2-4z+1\right)\)
\(=z^2\left(z^2-4z+1\right)+\left(4z^2-4z+1\right)\)
\(=z^2\left(z^2-4z+1\right)+\left[\left(2z\right)^2-2.2z.1+1^2\right]\)
\(=z^2\left(z-1\right)^2+\left(2z-1\right)^2\)
Ta có :
\(z^2\left(z-1\right)^2\ge0;\left(2z-1\right)^2\ge0\)
\(\Rightarrow z^2\left(z-1\right)^2+\left(2z-1\right)^2\ge0\) Dấu = xảy ra khi \(\left\{{}\begin{matrix}z-1=0\\2z-1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}z=1\\z=\dfrac{1}{2}\end{matrix}\right.\)
a, x.(x-y) +y.(x+y)
=x2-xy+xy+y2
=x2+y2
b, (x2-5).(2x+3)-2x.(x-3)
=2x3+3x2-10x-15-2x2+6x
=2x3-x2-4x-15
c, 8-5x.(x+2) +4 .( x-2) . (x+1) +2.( x+2)+ 2.(x-2)+10
=8-5x2-10x+4.(x2+x-2x-2)+2x+4+2x-4+10
=18-6x-5x2+4x2+4x-8x-8
=10-10x-x2
ks bn nobita nha