Find 4th degree polynomial that passes through $$(1, 2), (3, 5), (7, 8), (9, 10), (\sqrt2, \sqrt3)$$
I don't want to substitute those points into the polynomial which has $5$ unknowns. My approach was to let the polynomial be $f(x)$, and use the equation $f(x)=x+1$.
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How does $f(x) = x+1$ relate to anything?
Straightforward nonlinear data fit:
$$f(x) = 6.80 - 8.64 x + 4.54 x^2 - 0.73 x^3 + 0.037 x^4$$
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Write the five equations $$y_i=a+b\,x_i+c\,x_i^2+d\,x_i^3+e\,x_i^4$$
The last point is making trouble. So use the first four and solve for $(a,b,c,d)$ as a function of $e$. You should get $$a=189 e-\frac{5}{16}\quad b=\frac{127}{48}-300 e\quad c=130 e-\frac{17}{48}\quad d=\frac{1}{48}-20 e$$
Plug these in the fifth equation to get $$(453-340 \sqrt{2}) e+\frac{43}{8 \sqrt{2}}-\frac{49}{48}=\sqrt 3\implies e=\frac{49-129 \sqrt{2}+48 \sqrt{3}}{48 \left(453-340 \sqrt{2}\right)}$$ then $(a,b,c,d)$ and this is strictly exact.
Building upon your approach, note that if $f$ is the required quartic polynomial then $f(x)-x-1$ is a quartic polynomial with $1,7$ and $9$ as three of the roots. Hence, $$f(x)=x+1+(x-1)(x-7)(x-9)(ax+b)$$ Now, satisfy the remaining two points and solve to get the values of $a$ and $b$.