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Sửa đề: (sửa sai thì em làm lại:v) \(A=\left(x^2+3x+1\right)^2+\left(3x-1\right)^2-2\left(x^2+3x+1\right)\left(3x-1\right)\)
Đặt \(x^2+3x+1=a;3x-1=b\) cho nó dễ nhìn!
\(A=a^2+b^2-2ab=\left(a-b\right)^2\)
\(=\left(x^2+3x+1-\left(3x-1\right)\right)^2=\left(x^2+2\right)^2=x^4+4x^2+4\)
a/ \(x^3+1-x^2-x=\left(x+1\right)\left(x^2-x+1\right)-x\left(x+1\right)\)
\(=\left(x+1\right)\left(x^2-2x+1\right)=\left(x+1\right)\left(x-1\right)^2\)
b/ \(x^4-1-3\left(x^2+1\right)=\left(x^2+1\right)\left(x^2-1\right)-3\left(x^2+1\right)\)
\(=\left(x^2+1\right)\left(x^2-4\right)=\left(x^2+1\right)\left(x-2\right)\left(x+2\right)\)
c/ \(x^2+y^2-2xy-4z^2=\left(x-y\right)^2-\left(2z\right)^2\)
\(=\left(x-y-2z\right)\left(x-y+2z\right)\)
d/ \(x^2-4x+4-\left(y^2+6y+9\right)\)
\(=\left(x-2\right)^2-\left(y+3\right)^2\)
\(=\left(x+y+1\right)\left(x-y-5\right)\)
e/\(\left(x^2-2x+1\right)^3-\left(y^2\right)^3\)
\(=\left(x^2-2x+1-y^2\right)\left[\left(x^2-2x+1\right)^2+y^4+\left(x-1\right)^2y^2\right]\)
\(=\left[\left(x-1\right)^2-y^2\right]\left[\left(x-1\right)^4+y^4+2\left(x-1\right)^2y^2-\left(xy-y\right)^2\right]\)
\(=\left(x+y-1\right)\left(x-y-1\right)\left[\left(\left(x-1\right)^2+y^2\right)^2-\left(xy-y\right)^2\right]\)
\(=\left(x+y-1\right)\left(x-y-1\right)\left[\left(x-1\right)^2+y^2-xy+y\right]\left[\left(x-1\right)^2+y^2+xy-y\right]\)
f/ \(\left(x+y\right)^3-\left(x^3+y^3\right)\)
\(=x^3+y^3+3xy\left(x+y\right)-\left(x^3+y^3\right)\)
\(=3xy\left(x+y\right)\)
1. \(\left(x+1\right)^3-125\)
\(=\left(x+1\right)^3-5^3\)
\(=\left(x+1-5\right).\left[\left(x+1\right)^2+\left(x+1\right).5+5^2\right]\)
2. \(\left(x+4\right)^3-64\)
\(=\left(x+4\right)^3-4^3\)
\(=\left(x+4-4\right).\left[\left(x+4\right)^2+\left(x+4\right).4+4^2\right]\)
3. \(x^3-\left(y-1\right)^3\)
\(=(x^3-y+1).\left[\left(x^2\right)+x.\left(y+1\right)+\left(y+1\right)^2\right]\)
\(\)4. \(\left(a+b\right)^3-c^3\)
\(=\left[\left(a+b\right)-c\right].\left[\left(a+b\right)^2+\left(a+b\right).c+c^2\right]\)
5. \(125-\left(x+2\right)^3\)
\(=5^3-\left(x+2\right)^3\)
\(=\left(5-x-2\right).\left[5^2+5.\left(x+2\right)+\left(x+2\right)^2\right]\)
6. \(\left(x+1\right)^3+\left(x-2\right)^3\)
\(=\left[\left(x+1\right)+\left(x-2\right)\right].\left[\left(x+1\right)^2-\left(x+1\right).\left(x-2\right)+\left(x-2\right)^2\right]\)
Giải:
Gọi đa thức cần tìm là A
Ta có:
\(\dfrac{x^2}{x^2+7}=\dfrac{A}{x^4-49}\)
\(\Leftrightarrow\dfrac{x^2}{x^2+7}=\dfrac{A}{\left(x^7-7\right)\left(x^2+7\right)}\)
\(\Leftrightarrow\dfrac{x^2\left(x^2-7\right)}{\left(x^2+7\right)\left(x^2-7\right)}=\dfrac{A}{\left(x^2-7\right)\left(x^2+7\right)}\)
\(\Leftrightarrow x^2\left(x^2-7\right)=A\)
\(\Leftrightarrow A=x^4-7x^2\)
Vậy ...
Đáp án: C