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The segmentation of piecewise polynomial signals arises in a variety of scientific and engineering fields. When a signal is modeled as a piecewise polynomial, the key then becomes the detection of breakpoints followed by curve fitting and parameter estimation. This paper proposes HOPS, a fast High-Order Polynomial Segmenter, which is based on <inline-formula> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula>-penalized least-square regression. While the least-squares regression ensures fitting fidelity, the <inline-formula> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula> penalty takes the number of breakpoints into account. We show that dynamic programming can be applied to find the optimal solution to this problem and that a pruning strategy and matrix factorization can be utilized to accelerate the execution speed. Finally, we provide some illustrative examples, and compare the proposed method with state-of-the-art alternatives.