\(P=\dfrac{x^{10}-x^8+x^6-x^4+x^2-1}{x^4-1}\)
\(=\dfrac{\left(x^2-1\right)\left(x^8+x^4+1\right)}{\left(x^2-1\right)\left(x^2+1\right)}\)
\(=\dfrac{x^8+x^4+1}{x^2+1}\)
a) \(P=\dfrac{x^{10}-x^8+x^6-x^4+x^2-1}{x^4-1}=\dfrac{\left(x^2-1\right)\left(x^8+x^4+1\right)}{\left(x^2-1\right)\left(x^2+1\right)}=\dfrac{x^8+x^4+1}{x^2+1}\)
b) \(Q=\dfrac{x^{40}+x^{30}+x^{20}+x^{10}+1}{x^{45}+x^{40}+x^{35}+...+x^{10}+x^5+1}\)
\(\Rightarrow Q.x^5=\dfrac{x^{45}+x^{35}+x^{25}+x^{15}+x^5}{x^{45}+x^{40}+x^{35}+...+x^{10}+x^5+1}\)
\(\Rightarrow Q.x^5+Q=\dfrac{x^{45}+x^{40}+x^{35}+x^{30}+...+x^{10}+x^5+1}{x^{45}+x^{40}+x^{35}+...+x^{10}+x^5+1}=1\)
\(\Rightarrow Q\left(x^5-1\right)=1\Rightarrow Q=\dfrac{1}{x^5-1}\)
