## Quick Reference

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

*f*(*x*)=*a*_{n}*x*^{n}+*a*_{n−1}*x*^{n−1}+⋯+*a*_{1}*x*+*a*_{0},

*g*(*x*)=*b*_{n}*x*^{n}+*b*_{n−1}*x*^{n−1}+⋯+*b*_{1}*x*+*b*_{0},

where it is not necessarily assumed that *a*_{n} ≠ 0 and *b*_{n}≠ 0. If *f*(*x*)=*g*(*x*) for all values of *x*, then *a*_{n}=*b*_{n}, *a*_{n−1}=*b*_{n−1},…, *a*_{1}=*b*_{1}, *a*_{0}=*b*_{0}. 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

*x*^{3}=*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.