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Theo bài ra, ta có: \(x^2-y=y^2-x\Leftrightarrow x^2-y^2=-x+y\)
\(\Leftrightarrow\left(x-y\right)\left(x+y\right)=-\left(x-y\right)\)
\(\Leftrightarrow\left(x+y\right)=-1\)
Ta lại có: \(A=x^2+2xy+y^2-3x-3y=\left(x+y\right)^2-3\left(x+y\right)\)
Thay x+y=-1 vào biểu thức A, ta được: \(A=\left(-1\right)^2-3.\left(-1\right)=1+3=4\)
Vậy A=4
Câu 1: Ta có: A = \(x^3+y^3+3xy=x^3+y^3+3xy\times1=x^3+y^3+3xy\left(x+y\right)\)
\(=\left(x+y\right)^3=1^3=1\)
Câu 2: Ta có: \(B=x^3-y^3-3xy=\left(x-y\right)\left(x^2+xy+y^2\right)-3xy\)
\(=x^2+xy+y^2-3xy=x^2-2xy+y^2=\left(x-y\right)^2=1^2=1\)
Câu 3: Ta có: \(C=x^3+y^3+3xy\left(x^2+y^2\right)-6x^2.y^2\left(x+y\right)\)
\(=x^3+y^3+3xy\left(x^2+2xy+y^2-2xy\right)+6x^2y^2\)
\(=x^3+y^3+3xy\left(x+y\right)^2-3xy.2xy+6x^2y^2\)
\(=x^3+y^3+3xy.1-6x^2y^2+6x^2y^3\)
\(=x^3+y^3+3xy\left(x+y\right)=\left(x+y\right)^3=1^3=1\)
a, \(8^3yz+12^2yz+6xyz+yz\)
\(=512yz+144yz+6xyz+yz\)
\(=yz\left(512+14+6x+1\right)\)
\(=yz\left(527+6x\right)\)
$---$
b, \(81x^4\left(z^2-y^2\right)-z^2+y^2\)
\(=81x^4\left(z^2-y^2\right)-\left(z^2-y^2\right)\)
\(=\left(z^2-y^2\right)\left(81x^4-1\right)\)
\(=\left(z-y\right)\left(z+y\right)\left[\left(9x^2\right)^2-1^2\right]\)
\(=\left(z-y\right)\left(z+y\right)\left(9x^2-1\right)\left(9x^2+1\right)\)
\(=\left(z-y\right)\left(z+y\right)\left[\left(3x\right)^2-1^2\right]\left(9x^2+1\right)\)
\(=\left(z-y\right)\left(z+y\right)\left(3x-1\right)\left(3x+1\right)\left(9x^2+1\right)\)
$---$
c, \(\dfrac{x^3}{8}-\dfrac{y^3}{27}+\dfrac{x}{2}-\dfrac{y}{3}\)
\(=\left[\left(\dfrac{x}{2}\right)^3-\left(\dfrac{y}{3}\right)^3\right]+\left(\dfrac{x}{2}-\dfrac{y}{3}\right)\)
\(=\left(\dfrac{x}{2}-\dfrac{y}{3}\right)\left(\dfrac{x^2}{4}+\dfrac{xy}{6}+\dfrac{y^2}{9}\right)+\left(\dfrac{x}{2}-\dfrac{y}{3}\right)\)
\(=\left(\dfrac{x}{2}-\dfrac{y}{3}\right)\left(\dfrac{x^2}{4}+\dfrac{xy}{6}+\dfrac{y^2}{9}+1\right)\)
$---$
d, \(x^6+x^4+x^2y^2+y^4-y^6\)
\(=\left(x^6-y^6\right)+\left(x^4+x^2y^2+y^4\right)\)
\(=\left[\left(x^2\right)^3-\left(y^2\right)^3\right]+\left(x^4+x^2y^2+y^4\right)\)
\(=\left(x^2-y^2\right)\left(x^4+x^2y^2+y^4\right)+\left(x^4+x^2y^2+y^4\right)\)
\(=\left(x^4+x^2y^2+y^4\right)\left(x^2-y^2+1\right)\)
$Toru$
Bài 3:
\(\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+3\right)+15\)
\(=\left(x^2-9\right)\left(x^2-1\right)+15\)
\(=x^4-10x^2+9+15\)
\(=x^4-10x^2+24\)
\(=\left(x^2-4\right)\left(x^2-6\right)\)
\(=\left(x-2\right)\left(x+2\right)\left(x^2-6\right)\)
Ta có a + b = 3
=> (a + b)2 = 9
=> a2 + 2ab + b2 = 9
=> a2 + b2 = 5 (ab = 2)
Khi a2 + b2 = 5 => a2 - 2ab + b2 = 1
=> (a - b)2 = 1
=> a - b = \(\pm1\)
Đặt A \(\frac{1}{a^3}-\frac{1}{b^3}=\frac{b^3-a^3}{\left(a.b\right)^3}=\frac{\left(b-a\right)\left(b^2+ab+a^2\right)}{\left(ab\right)^3}=-\frac{\left(a-b\right)\left(a^2+ab+b^2\right)}{\left(ab\right)^3}\)
Với a - b = 1 ; ab = 2 ; a2 + b2 = 5 ta có A = \(-\frac{1.\left(5+2\right)}{2^3}=-\frac{7}{8}\)
Với a - b = - 1 ; ab = 2 ; a2 + b2 = 5 ta có A = \(-\frac{\left(-1\right).\left(5+2\right)}{2^3}=\frac{7}{8}\)
Ta có: \(\hept{\begin{cases}a+b=3\\ab=2\end{cases}}\Leftrightarrow\hept{\begin{cases}\left(a+b\right)^2=9\\ab=2\end{cases}\Leftrightarrow}\hept{\begin{cases}a^2+2ab+b^2=9\\ab=2\end{cases}}\Leftrightarrow\hept{\begin{cases}a^2+b^2=5\\ab=2\end{cases}}\)
Khi đó: \(\frac{1}{a^3}-\frac{1}{b^3}=\frac{b^3-a^3}{a^3b^3}=\frac{\left(b-a\right)\left(a^2+ab+b^2\right)}{8}=\frac{7\left(b-a\right)}{8}\)
Ta có: \(a+b=3\Rightarrow a=3-b\) thay vào: \(\left(3-b\right)b=2\)
\(\Leftrightarrow b^2-3b+2=0\Leftrightarrow\left(b-1\right)\left(b-2\right)=0\Leftrightarrow\orbr{\begin{cases}b=1\Rightarrow a=2\\b=2\Rightarrow a=1\end{cases}}\)
Nếu \(\hept{\begin{cases}a=2\\b=1\end{cases}\Rightarrow}\frac{1}{a^3}-\frac{1}{b^3}=-\frac{7}{8}\)
Nếu \(\hept{\begin{cases}a=1\\b=2\end{cases}}\Rightarrow\frac{1}{a^3}-\frac{1}{b^3}=\frac{7}{8}\)