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Substitution is a basic operation in algebra, in particular in computer algebra. [10] [11] A common case of substitution involves polynomials, where substitution of a numerical value (or another expression) for the indeterminate of a univariate polynomial amounts to evaluating the polynomial at that value.
Because of this, often, the only simple effective way to deal with multiplication by expressions involving variables is to substitute each of the solutions obtained into the original equation and confirm that this yields a valid equation. After discarding solutions that yield an invalid equation, we will have the correct set of solutions.
The conditions on the theorem can be weakened in various ways. First, the requirement that φ be continuously differentiable can be replaced by the weaker assumption that φ be merely differentiable and have a continuous inverse. [4] This is guaranteed to hold if φ is continuously differentiable by the inverse function theorem.
Substitution (algebra), replacing occurrences of some symbol by a given value; Substitution (logic), a syntactic transformation on strings of symbols of a formal language; String substitution, a mapping of letters in an alphabet to languages
A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [14]
In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials [1] —hence the method may be referred to as the FOIL method.The word FOIL is an acronym for the four terms of the product:
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