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equating coefficients

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Let f(x) and g(x) be polynomials, and let

f(x)=anxn+an−1xn−1+⋯+a1x+a0,

g(x)=bnxn+bn−1xn−1+⋯+b1x+b0,

where it is not necessarily assumed that an ≠ 0 and bn≠ 0. If f(x)=g(x) for all values of x, then an=bn, an−1=bn−1,…, a1=b1, a0=b0. Using this fact is known as equating coefficients. The result is obtained by applying the Fundamental Theorem of Algebra to the polynomial h(x), where h(x)=f(x)−g(x). If h(x)=0 for all values of x (or, indeed, for more than n values of x), the only possibility is that h(x) is the zero polynomial with all its coefficients zero. The method can be used, for example, to find numbers A, B, C and D such that

x3=A(x−1)(x−2)(x−3)+B(x−1)(x−2)+C(x−1)+D

for all values of x. It is often used to find the unknowns in partial fractions.

Subjects: Mathematics.

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