@whuber has given a really excellent answer here. I just want to add a small complimentary point. The question states that "a linear relation of predictor and data is not interpretable". This hints at a common misunderstanding, although I usually hear it on the other end ('what is the interpretation of the squared [cubic, etc.] term?').
When we have a model with multiple different covariates, each beta [term] can generally be afforded its own interpretation. For example, if:
$$
\widehat{\text{GPA}}_{college}=\beta_0+\beta_1\text{GPA}_{highschool}+\beta_2\text{class rank}+\beta_3\text{SAT},
$$
(GPA means grade point average;
rank is the ordering of a student's GPA relative to other students at the same high school; &
SAT means 'scholastic aptitude test' a standard, nationwide test for students going to university)
then we can assign separate interpretations to each beta/term. For instance, if a student's high school GPA were 1 point higher--all else being equal--we would expect their college GPA to be $\beta_1$ points higher.
It is important to note, however, that it is not always permissible to interpret a model in this manner. One obvious case is when there is an interaction amongst some of the variables, as it would not be possible for the individual term to differ and still have all else held constant--of necessity, the interaction term would change as well. Thus, when there is an interaction, we do not interpret main effects but only simple effects, as is well understood.
The situation with power terms is directly analogous, but unfortunately, does not seem to be widely understood. Consider the following model:
$$
\hat{y}=\beta_0+\beta_1x+\beta_2x^2
$$
(In this situation, $x$ is intended to represent a prototypical continuous covariate.) It is not possible for $x$ to change without $x^2$ changing also, and vice versa. Simply put, when there are polynomial terms in a model, the various terms based on the same underlying covariate are not afforded separate interpretations. The $x^2$ ($x$, $x^{17}$, etc.) term does not have any independent meaning. The fact that a $p$-power polynomial term is 'significant' in a model indicates that there are $p-1$ 'bends' in the function relating $x$ and $y$. It is unfortunate, but unavoidable, that when curvature exists, the interpretation becomes more complicated, and possibly less intuitive. To assess the change in $\hat{y}$ as $x$ changes, we will have to use calculus. The derivative of the above model is:
$$
\frac{dy}{dx}=\beta_1+2\beta_2x
$$
which is the instantaneous rate of change in the expected value of $y$ as $x$ changes, all else being equal. This is not so clean as the interpretation of the very top model; importantly, the instantaneous rate of change in $y$ depends on the level of $x$ from which the change is assessed. Furthermore, the rate of change in $y$ is an instantaneous rate; that is, it is itself continuously changing throughout the interval from $x_{old}$ to $x_{new}$. This is simply the nature of a curvilinear relationship.
