# Interpreting multiple polynomial regression coefficients

I read a couple post on interpreting polynomial coefficients here in cross validate however none of them touch on how to interpret multiple polynomial regression coefficients. Perhaps its the same but I wanted to ask the question for my own edification as well as others who may be wondering.

Here is a regression I just ran where there are four terms each with a corresponding polynomial term. How would one go about interpreting this output?

Call:
lm(formula = a ~ t + d + r + p + I(t^2) +
I(d^2) + I(r^2) + I(p^2), data = df)

Residuals:
Min      1Q  Median      3Q     Max
-3.8466 -1.4200 -0.2556  1.8784  6.9382

Coefficients:
Estimate         Std. Error t value Pr(>|t|)
(Intercept)   1.071213896096506  0.897660289412562   1.193 0.244901
t            -0.000016729186474  0.000012896669665  -1.297 0.207434
d             0.000240787673662  0.000136472581690   1.764 0.090949 .
r             0.000936217403829  0.000238344538301   3.928 0.000673 ***
p            -0.000410104711084  0.000260680628526  -1.573 0.129327
I(t^2)        0.000000000005504  0.000000000024388   0.226 0.823423
I(d^2)       -0.000000000948744  0.000000002529495  -0.375 0.711043
I(r^2)       -0.000000006440508  0.000000002136199  -3.015 0.006170 **
I(p^2)        0.000000007091433  0.000000007243474   0.979 0.337761
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.893 on 23 degrees of freedom
Multiple R-squared:  0.8754,    Adjusted R-squared:  0.832
F-statistic:  20.2 on 8 and 23 DF,  p-value: 0.00000001125


The first thing I would do is rescale the independent variables so there are fewer leading zeroes after the decimal. Maybe multiply each by 1000.