Local polynomial coefficients without differentiation
The coefficient of z^j is the number multiplying that power after terms are collected. Write [z^j]P for it. To expand a polynomial H near a rational number t, substitute T=t+z and multiply out. For H(T)=2+3T+T^2 and t=1, H(1+z)=6+5z+z^2. These are exact polynomial identities, not numerical approximations or infinite series. For a positive integer J, keeping just powers below J is written modulo z^J; for J=2 the stored list is (6,5). Add and multiply such lists while dropping powers J and above: nonnegative powers cannot later contribute to a lower power. These local coefficients are often called Taylor coefficients; no derivatives are needed here.
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A data analyst uses the polynomial expression 8x^2y^4 - 3x^3y + 12 to model the efficiency of a processing plant. Based on the definition of a polynomial's degree, what is the degree of this expression?
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