IDA: Example

Examples

5 + X - 2X² has (nonzero) terms 5, X, - 2X². The coefficients of the polynomial are 5, 1, -2. The monomials are 1 (= X⁰), X, X².
Instead of 2 + 3X + 0X² - 5X³ we write 2 + 3X - 5X³.
In (Z/6Z)[X], the polynomials 2X² , 2X² - 6X⁴ and 0 + 12X + 8X² - 24X³ + 300X⁹ are equal.
4 - Y + 7Y³ is a polynomial in Y.
The expression
1/(X + 1)
is not a polynomial.
An important polynomial in linear algebra is the characteristic polynomial of a square n by n matrix A. It plays an essential role in the analysis of matrices and linear maps. By definition it is the following polynomial in X:
det(XI - A),
where I is the n by n identity matrix and where det stands for the determinant. To evaluate such a determinant, you need to know how to do polynomial arithmetic. Further discussion in another Example, when the arithmetic of polynomials is explained. See also the Prerequisites.