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a)
\(P=\dfrac{x^{10}-x^8+x^6-x^4+x^2-1}{x^4-1}\)
\(=\dfrac{x^8\left(x^2-1\right)+x^4\left(x^2-1\right)+\left(x^2-1\right)}{\left(x^2-1\right)\left(x^2+1\right)}\)
\(=\dfrac{\left(x^2-1\right)\left(x^8+x^4+1\right)}{\left(x^2-1\right)\left(x^2+1\right)}\)
\(=\dfrac{x^8+x^4+1}{x^2+1}\)
b)
\(Q=\dfrac{x^{40}+x^{30}+x^{20}+x^{10}+1}{x^{45}+x^{40}+x^{35}+...+x^{10}+x^5+1}\)
\(=\dfrac{x^{40}+x^{30}+x^{20}+x^{10}+1}{\left(x^{45}+x^{35}+...+x^5\right)+\left(x^{40}+x^{30}+...+1\right)}\)
\(=\dfrac{x^{40}+x^{30}+x^{20}+x^{10}+1}{x^5\left(x^{40}+x^{30}+...+1\right)+\left(x^{40}+x^{30}+...+1\right)}\)
\(=\dfrac{x^{40}+x^{30}+x^{20}+x^{10}+1}{\left(x^{40}+x^{30}+...+1\right)\left(x^5+1\right)}\)
\(=\dfrac{1}{\left(x^5+1\right)}\)
cái câu b dòng cuối mẫu số đóng mở ngoặc chi cho mệt ei =.=
Đặt biểu thức là A, ta có:
\(A=\frac{x^{40}+x^{30}+x^{20}+x^{10}+1}{x^{45}+x^{40}+x^{35}+...+x^{10}+x^5+1}\)
\(\Rightarrow A.x^5=\frac{x^{45}+x^{35}+x^{25}+x^{15}+x^5}{x^{45}+x^{40}+x^{35}+...+x^{10}+x^5+1}\)
\(\Rightarrow A.x^5+A=\frac{x^{45}+x^{40}+x^{35}+x^{25}+x^{15}+x^5+x^{40}+x^{30}+x^{20}+x^{10}+1}{x^{45}+x^{40}+x^{35}+...+x^{10}+x^5+1}\)
\(\Rightarrow A.x^5+1=1\)
\(\Rightarrow A=\frac{1}{x^5+1}\)
\(\dfrac{1}{x^2+7x+10}+\dfrac{1}{x^2+13x+40}+\dfrac{1}{x^2+19x+88}+\dfrac{1}{x^2+25x+154}\)
\(=\dfrac{1}{\left(x+2\right)\left(x+5\right)}+\dfrac{1}{\left(x+5\right)\left(x+8\right)}+\dfrac{1}{\left(x+8\right)\left(x+11\right)}+\dfrac{1}{\left(x+11\right)\left(x+14\right)}\)
\(=\dfrac{1}{x+2}-\dfrac{1}{x+5}+\dfrac{1}{x+5}-\dfrac{1}{x+8}+\dfrac{1}{x+8}-\dfrac{1}{x+11}+\dfrac{1}{x+11}-\dfrac{1}{x+14}\)
\(=\dfrac{1}{x+2}-\dfrac{1}{x+14}\)
\(\dfrac{x-55}{45}+\dfrac{x-30}{35}+\dfrac{x-25}{25}+\dfrac{x-40}{15}=10\)
\(< =>\dfrac{x-55}{45}+\dfrac{x-30}{35}+\dfrac{x-25}{25}+\dfrac{x-40}{15}-10=0\)
\(< =>\dfrac{x-55}{45}-1+\dfrac{x-30}{35}-2+\dfrac{x-25}{25}-3+\dfrac{x-40}{15}-4=0\)
\(< =>\dfrac{x-100}{45}+\dfrac{x-100}{35}+\dfrac{x-100}{25}+\dfrac{x-100}{15}=0\)
\(< =>\left(x-100\right)\left(\dfrac{1}{45}+\dfrac{1}{35}+\dfrac{1}{25}+\dfrac{1}{15}\right)=0\)
\(< =>x-100=0\left(\dfrac{1}{45}+\dfrac{1}{35}+\dfrac{1}{25}+\dfrac{1}{15}\ne0\right)\)
\(< =>x=100\)