a: \(f\left(x\right)=1+x+x^2+...+x^{2021}+x^{2022}\)
\(f\left(0\right)=1+0+0^2+...+0^{2021}+0^{2022}=1\)
\(f\left(1\right)=1+1+1^2+...+1^{2021}+1^{2022}\)
\(=1+1+...+1\)
=2023
\(f\left(-1\right)=1+\left(-1\right)+\left(-1\right)^2+\left(-1\right)^3+...+\left(-1\right)^{2021}+\left(-1\right)^{2022}\)
\(=\left(1-1\right)+\left(1-1\right)+...+\left(1-1\right)+1\)
=1
b: \(g\left(x\right)=100x^{100}+99x^{99}+...+2x^2+x+1\)
=>\(g\left(1\right)=100\cdot1^{100}+99\cdot1^{99}+...+2\cdot1^2+1+1\)
\(=100+99+...+2+1+1\)
\(=\left(1+2+...+99+100\right)+1\)
\(=\dfrac{100\cdot101}{2}+1=5050+1=5051\)
c: \(h\left(x\right)=1-x+x^2-x^3+...+x^{100}\)
\(=1-\left(-1\right)+\left(-1\right)^2-\left(-1\right)^3+...+\left(-1\right)^{100}\)
=1+1+...+1
=101
