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\(x^2-x+1-m=0\left(1\right)\\ \text{PT có 2 nghiệm }x_1,x_2\\ \Leftrightarrow\Delta=1-4\left(1-m\right)\ge0\\ \Leftrightarrow4m-3\ge0\Leftrightarrow m\ge\dfrac{3}{4}\\ \text{Vi-ét: }\left\{{}\begin{matrix}x_1+x_2=1\\x_1x_2=1-m\end{matrix}\right.\\ \text{Ta có }5\left(\dfrac{1}{x_1}+\dfrac{1}{x_2}\right)-x_1x_2+4=0\\ \Leftrightarrow5\cdot\dfrac{x_1+x_2}{x_1x_2}-x_1x_2+4=0\\ \Leftrightarrow\dfrac{5}{1-m}+m-1+4=0\\ \Leftrightarrow\dfrac{5}{1-m}+m+3=0\\ \Leftrightarrow5+\left(1-m\right)\left(m+3\right)=0\\ \Leftrightarrow m^2+2m-8=0\\ \Leftrightarrow m^2-2m+4m-8=0\\ \Leftrightarrow\left(m-2\right)\left(m+4\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}m=2\left(n\right)\\m=-4\left(l\right)\end{matrix}\right.\)
Vậy $m=2$
a,\(\left(x^2+x\right)2+3\left(x^2+x\right)+2\)
=\(\left(x^2+x\right)6+2\)
b,\(\left(x^2+x\right)2-2\left(x^2+x\right)-15\)
=\(-4\left(x^2+x\right)-15\)
c,\(\left(x^2+x+1\right)\left(x^2+x+2\right)-12\)
=\(\left(x^2+x+1\right)\left(x^2+x+1\right)+1-12\)
=\(\left(x^2+x+1\right)^2-11\)
d,\(\left(x^2+x\right)2+4x^2+4x-12\)
=\(x\left(x+1\right)2+2x\left(x+1\right)-12\)
=\(2x\left(x+1\right)+2x\left(x+1\right)-12\)
=\(\left(x+1\right)\left(2x+2x-12\right)\)
= \(\left(x+1\right)\left(4x-12\right)=4\left(x+1\right)\left(x-3\right)\)
e,\(\left(x^2+2x\right)2+9x^2+18x+20\)
=\(x\left(x+2\right)2+9x\left(x+2\right)+20\)
=\(2x\left(x+2\right)+9x\left(x+2\right)+20=\left(x+2\right)\left(2x+9x+20\right)\)
=\(\left(x+2\right)\left(11x+20\right)\)
Đặt \(t=x^2\left(t\ge0\right)\)
Khi đó phương trình ban đầu tương đương với pt\(t^2-2\left(m+2\right)t+m^2-2m+3=0\) (*)
Để pt ban đầu có 4 nghiệm phân biệt thì pt (*) có hai nghiệm dương phân biệt ⇔
\(\left\{{}\begin{matrix}\Delta>0\\S>0\\P>0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(m+2\right)^2-m^2+2m-3>0\\2\left(m+2\right)>0\\m^2-2m+3>0\end{matrix}\right.\)
⇔ \(\left\{{}\begin{matrix}6m+1>0\\m+2>0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}m>-\dfrac{1}{6}\\m>-2\end{matrix}\right.\)
⇔ \(m>-\dfrac{1}{6}.\)
Giả sử (*) có hai nghiệm là t1, t2. Khi đó theo Viet ta có t1.t2 = m2 - 2m + 3.
Ta có: x1.x2.x3.x4 = t1.t2 = m2 - 2m +3.
Ta có E = m2 - 2m + 3 = (m - 1)2 + 2 ≥ 2.
Min E = 2. Dấu bằng xảy ra khi m = 1.
a) Ta có: \(x^4+4\)
\(=\left(x^2\right)^2+2\cdot x^2\cdot2+4-4x^2\)
\(=\left(x^2+2\right)^2-\left(2x\right)^2\)
\(=\left(x^2-2x+2\right)\left(x^2+2x+2\right)\)
b) Ta có: \(x^4+64\)
\(=\left(x^2\right)^2+8^2+16x^2-16x^2\)
\(=\left(x^2+8\right)^2-\left(4x\right)^2\)
\(=\left(x^2-4x+8\right)\left(x^2+4x+8\right)\)
c) Ta có: \(64x^4+y^4\)
\(=\left(8x^2\right)^2+\left(y^2\right)^2+16x^2y^2-16x^2y^2\)
\(=\left(8x^2+y^2\right)^2-\left(4xy\right)^2\)
\(=\left(8x^2-4xy+y^2\right)\left(8x^2+4xy+y^2\right)\)
d) Ta có: \(x^3-x^2-4\)
\(=x^3+x^2+2x-2x^2-2x-4\)
\(=x\left(x^2+x+2\right)-2\left(x^2+x+2\right)\)
\(=\left(x^2+x+2\right)\left(x-2\right)\)
e) Ta có: \(x^3-7x-6\)
\(=x^3-x-6x-6\)
\(=x\left(x^2-1\right)-6\left(x+1\right)\)
\(=x\left(x+1\right)\left(x-1\right)-6\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2-x-6\right)\)
\(=\left(x+1\right)\left(x^2-3x+2x-6\right)\)
\(=\left(x+1\right)\left(x-3\right)\left(x+2\right)\)
f) Ta có: \(x^4+x^2+1\)
\(=x^4+2x^2+1-x^2\)
\(=\left(x^2+1\right)^2-x^2\)
\(=\left(x^2-x+1\right)\left(x^2+x+1\right)\)