It is asked to find a polynomial of adequate degree to estimate $\sqrt{1.035}$. The following table is given:
We know that 1.03 and 1.04 need to be used. From the divided differences table, we see that they are increasing, and we also have some zeros in the second-order divided differences.
I don't get what the author means by "adequate degree". For example, if we start in 1.02, we only can interpolate by a line. But I cannot decide if I should start in 1.01 or 1.
The first order differences are almost constant here, they decrease very slowly. The simple linear interpolation between $1.03$ and $1.04$ is appropriate. It gives $$ \sqrt{1.035}\approx \frac{1.0149+1.0198}{2} = 1.01735 $$
Comparing this to $\sqrt{1.035} = 1.017349497\ldots$ confirms there is no need for higher order interpolation. The result has as many correct significant digits as the input $1.035$.
Another sign that higher degrees are not useful: quadratic interpolation gives a polynomial with leading coefficient about $10^{-12}$.