There were, and still are, two main tasks that are closely related mathematically but that occur in different practical situations.

(1) Given two lists of n numbers, where all the x's are different

x_{1}, x_{2}, ..., x_{n}

y_{1},
y_{2}, ..., y_{n}

find a polynomial, P(x), of degree n or less, such that

P(x_{1}) = y_{1}, P(x_{2})
= y_{2}, ..., P(x_{n}) = y_{n}.

The construction of such "interpolating" polynomials came from numerical calculations, which are required in many applications of calculus. It can be done by solving a system of n linear equations with n unknowns. (The unknowns are the coefficients of the polynomial that is being constructed.) Before the invention of computers, solving "large" systems of linear equations was very difficult, so other methods of construction were designed. The two best known were the methods of Newton and of Lagrange.

At present you can easily solve such systems of equations up to n = 10 on the TI-83/84. Theoretically you could solve systems of n linear equations with n unknowns on this calculator for n = 33, but in practice, large systems cannot be solved because of rounding errors.

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