Calculating Adjusted $R^2$ in Polynomial Linear Regression with Single Variable

When calculating Adjusted $R^2$ the formula is $1-(1-R^2)\frac{n-1}{n-k-1}$ with $k$ being how many predictors you have. If I am using a model with a single variable but that variable has been put to the 4th, 3rd, and 2nd power like the following,

$\hat{Y}=-0.0162x^4+0.2239x^3-1.0941x^2+2.0972x-0.9513$

would I have a single predictor or would I count each powered term as a predictor? Also if you could give short reasoning so I can try and grasp the concept as to why or why not to.

Yes, each term counts as a predictor. What you're doing is fitting a model with 5 parameters (4 plus the intercept). To make this more clear, as far as the fitting is concerned you have constructed a new set of variables $x_1 = x$, $x_2 = x^2$, ... $x_4 = x^4$ and you have fitted a regression with $x_1$ to $x_4$ as predictors.
• Thank you Glen, as a counter question is the intercept counted as a predictor with Adjusted $R^2$? – DanTheMan Oct 24 '12 at 21:51