a) \(A=x^2+x+1=x^2+x+\frac{1}{4}+\frac{3}{4}=\left(x+\frac{1}{2}\right)^2+\frac{3}{4}>0,\forall x\).

b) \(B=x^2-xy+y^2=x^2-xy+\frac{1}{4}y^2+\frac{3}{4}y^2=\left(x-\frac{1}{2}y\right)^2+\frac{3}{4}y^2\ge0\)

Dấu \(=\)khi \(\hept{\begin{cases}x-\frac{1}{2}y=0\\y=0\end{cases}}\Leftrightarrow x=y=0\).

c) \(C=4x-10-x^2=-\left(x^2-4x+4\right)-6=-\left(x-2\right)^2-6< 0,\forall x\).

a) \(x^2+x+1\)

\(=x^2+x+\frac{1}{4}+\frac{3}{4}\)

\(=\left(x+\frac{1}{2}\right)^2+\frac{3}{4}>0\)

Ta có đpcm

P=x²y²+y²z²+x²z²+2xy²z+2xyz²+2x²yz+x⁴+y²z²-2x²yz+y⁴+x²z²-2xy²x+z⁴+x²y²-2xyx²=

=x⁴+y⁴+z⁴+2x²y²+2y²z²+2x²z²=(x²+y²+z²)²=10²=100

= (ax+a-ay) + (by-bx-b)

= a nhân ( x+1-y) + b nhân ( y-x-1 )

= a nhân ( x+1-y) - b nhân ( x+1-y )          

= (x+1-y) nhân (a-b) 

Các bạn ơi giúp mình với. Làm ơn, mình gấp lắm

\(x^2+y^2\ge5\left(x-y\right)^2\Leftrightarrow4x^2-10xy+4y^2\le0\)

\(\Leftrightarrow\left(4x-2y\right)\left(x-2y\right)\le0\)

\(\Rightarrow\frac{1}{2}\le\frac{x}{y}\le2\)

Bài 3  Tính nhanh

A, 892^2+892.216+108^2                 B, 36^2+26^2-52.36

  =892^2+2.892.108+108^2                 =36^2-52.62+26^2

  =(892+108)^2           

  =1000^2

  =1000000

Bài 4 Phân tích đa thức sau thành nhân tử   

X^3-2x^2+x

5(x-y)-y(x-y)

36-12x+x^2

4x^2+12x-9

Bài 3:

\(892^2+892.216+108^2=892^2+2.892.108+108^2=\left(892+108\right)^2=1000000\)

\(36^2+26^2-52.36=36^2-2.26.36+26^2=\left(36-26\right)^2=100\)

Bài 4: 

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

\(5.\left(x-y\right)-y.\left(x-y\right)=\left(5-y\right)\left(x-y\right)\)

\(36-12x+x^2=x^2-12x+36=x^2-2x.6+6^2=\left(x-6\right)^2\)

\(4x^2+12x-9=\left(2x\right)^2+2.2x.3+3^2=\left(2x+3\right)^2\)

\(\left(5x-y\right)^2\)

\(=25x^2-10xy+y^2\)

\(\left(5x-y\right)^2\)

\(=25x^2-10xy+y^2\)

= (2x^2)^2 -2 nhân 2x^2 nhân 2x + (2x)^2 

= (2x^2 - 2x) 

\(a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)\)

\(=-c\left(a^2-ab+b^2\right)\)

\(=-ca^2-b^2c+abc\)

Ta có đpcm

\(x^2+5x+6\)

\(=x^2+2x+3x+6\)

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

\(=x\left(x+2\right)+3\left(x+2\right)\)

\(=\left(x+2\right)\left(x+3\right)\)

\(x^2+5x+6\)

\(=x^2+2x+3x+6\)

\(=\left(x+2\right)\left(x+3\right)\)