In this work, I develop a systematic approach to classifying multipartite entanglement. I show that the response of a quantum state to the action of operators of different sizes is related to its entanglement characteristic. Such responses can be described by a polynomial which I call entanglement polynomial. The entanglement polynomial induces a monoid homomorphism from quantum states to polynomials. By taking the quotient over the kernel of the homomorphism, we obtain an isomorphism from entanglement classes to polynomials, which classifies entanglement effectively. It implies that we can characterize and find the building blocks of entanglement by polynomial factorization. It can be shown that the entanglement polynomial is a SLOCC invariant. To calculate the entanglement polynomial practically, I construct a series of states, called renormalized states, whose ranks are related to the coefficients of the entanglement polynomial.
