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\(8^{13}-9.8^{12}+9.8^{11}-9.8^{10}+.....-9.8^2+9.8-2\)
\(=8^{13}-\left(8+1\right).8^{12}+\left(8+1\right).8^{11}-\left(8+1\right).8^{10}+....-\left(8+1\right).8^2+\left(8+1\right).8-2\)
\(=8^{13}-8^{13}-8^{12}+8^{12}+8^{11}-8^{11}-8^{10}+....-8^3-8^2+8^2+8-2\)
\(=\left(8^{13}-8^{13}\right)-\left(8^{12}-8^{12}\right)+\left(8^{11}-8^{11}\right)-....-\left(8^2-8^2\right)+8-2\)
\(=8-2=6\)
x=8 nên x+1=9
\(F=x^{13}-9x^{12}+9x^{11}-9x^{10}+...-9x^2+9x-2\)
\(=x^{13}-x^{12}\left(x+1\right)+x^{11}\left(x+1\right)-x^{10}\left(x+1\right)+...-x^2\left(x+1\right)+x\left(x+1\right)-2\)
\(=x^{13}-x^{13}-x^{12}+x^{12}+...-x^3-x^2+x^2+x-2\)
=x-2
=8-2
=6
Với x = 8
=> x + 1 = 9 (1)
Thay (1) vào biểu thức ta được
\(x^{10}-9x^9+9x^8-9x^7+...+9x^2-9x-2\)
\(=x^{10}-\left(x+1\right)x^9+\left(x+1\right)x^8-\left(x+1\right)x^7+...+\left(x+1\right)x^2-\left(x+1\right)x-2\)
\(=x^{10}-x^{10}-x^9+x^9+x^8-x^8-x^7+...+x^3+x^2-x^2-x-2\)
\(=-x-2\)
\(=-8-2=-10\)
\(C=\left(x^2+x+1\right)\left(x^2+x+2\right)-12\)
Đặt \(x^2+x+1=t\)
Ta được:
\(C=t\left(t+1\right)-12\)
\(C=t^2+t-12\)
\(C=t^2+4t-3t-12\)
\(C=t\left(t+4\right)-3\left(t+4\right)\)
\(C=\left(t+4\right)\left(x-3\right)\)
\(C=\left(x^2+x+5\right)\left(x^2+x-2\right)\)
\(C=\left(x^2+x+5\right)\left(x^2-x+2x-2\right)\)
\(C=\left(x^2+x+5\right)\left[x\left(x-1\right)+2\left(x-1\right)\right]\)
\(C=\left(x^2+x+5\right)\left(x-1\right)\left(x+2\right)\)
Vậy....
\(\left(x^2+x\right)^2+9x^2+9x+14\)
\(=x^4+2x^3+10x^2+9x+14\)
\(=x^4+x^3+2x^2+x^3+x^2+2x+7x^2+7x+14\)
\(=x^2\left(x^2+x+2\right)+x\left(x^2+x+2\right)+7\left(x^2+x+2\right) \)
\(=\left(x^2+x+2\right)\left(x^2+x+7\right)\)