ABSTRACT
Two polynomials are equal if they have the same coefficients of like powers of the variable.
5.1.1-2. Main operations over polynomials.
1◦. The sum (difference) of two polynomials f (x) of degree n and g(x) of degree m is the polynomial of degree l ≤ max{n, m} whose coefficient of each power of x is equal to the sum (difference) of the coefficients of the same power of x in f (x) and g(x), i.e. if
g(x) ≡ bmxm + bm-1xm-1 + · · · + b1x + b0, (5.1.1.2) then the sum (difference) of polynomials (5.1.1.1) and (5.1.1.2) is
f (x) g(x) = clx l + cl-1x
l-1 + · · · + c1x + c0, where ck = ak bk (k = 0, 1, . . . , l). If n > m then bm+1 = . . . = bn = 0; if n < m then an+1 = . . . = am = 0. 2◦. To multiply a polynomial f (x) of degree n by a polynomial g(x) of degree m, one should multiply each term in f (x) by each term in g(x), add the products, and collect similar terms. The degree of the resulting polynomial is n+m. The product of polynomials (5.1.1.1) and (5.1.1.2) is
f (x)g(x) = cn+mx n+m + cn+m-1x
n+m-1 + · · · + c1x + c0, ck = i+j=k∑
aibj ,
where k = 0, 1, . . . , n + m. 3◦. Each polynomial f (x) of degree n can be divided by any other polynomial p(x) of degree m (p(x) ≠ 0) with remainder, i.e., uniquely represented in the form f (x) = p(x)q(x)+ r(x), where q(x) is a polynomial of degree n – m (for m ≤ n) or q(x) = 0 (for m > n), referred to as the quotient, and r(x) is a polynomial of degree l < m or r(x) = 0, referred to as the remainder.
