Examples
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The sum of 5 + X - 2X2 and -3 - X is
(5 - 3) + (1 - 1)X + (-2 + 0)X2 = 2 - 2 X2. -
The product of 5 + X - 2X2 and 3 + X is the polynomial
(5 · 3) + (5 · 1 + 1 · 3)X + (1 · 1 + (-2) · 3)X2 + ((-2) · 1)X3 = 15 + 8X - 5X2 - 2X3.
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In Q[X] we have
(X + 1)2 = X2 + 2X + 1. In Z/2Z[X] we have
(X + 1)2 = X2 + 1, since 2 = 0 and therefore 2X = 0.
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When we expand the parentheses in the expression
(9X4)3 + (3X - 9X4)3 - (9X3 - 1)3
and collect equal powers of X, we obtain the expression 1. We conclude that this polynomial is equal to 1. Here, `expansion' is based on the operations `multiplication and addition of polynomials'. An interpretation of the equality is that, for every integer one substitutes for X, the resulting number is 1.
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Similarly, we have
(X2 - 1)2 + (2X)2 = (X2 + 1)2.
Even more so than for integers, different notations for the same expression are possible. -
The set of polynomials Q[X] forms a vector space
over Q (scalar multiplication with q
Q is defined as multiplication with the constant polynomial
q). This vector space is infinite-dimensional. Now let m be a
positive integer. The polynomials in Q[X] of degree
< m form a finite-dimensional vector space of dimension m.
The m monomials 1, X, ..., Xm - 1 are a basis.
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The characteristic polynomial of the 2 by 2 matrix A = [[2, 3], [1, 5]]
is
det(XI - A) = (X - 2)(X - 5) - 3 · 1 = X2 - 7X + 7. (See the Prerequisites.)