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Q3 Why are polynomials closed under multiplication?
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Let f(x) = a_0 + a_1 x + ... + a_m x^m and g(x) = b_0 + b_1 x + ... + b_n x^n be polynomials over a ring (or field) R. Their product is
f(x)g(x) = sum_{i=0}^m sum_{j=0}^n a_i b_j x^{i+j}.
Collecting like powers x^k (where k = i + j) gives a finite polynomial
f(x)g(x) = c_0 + c_1 x + ... + c_{m+n} x^{m+n},
with coefficients
c_k = sum_{i=0}^k a_i b_{k-i},
where terms with indices outside the ranges of a_i or b_j are taken as zero. Each c_k is a finite sum in R, so f(x)g(x) is again a polynomial in R[x].
Thus polynomials are closed under multiplication. (Also deg(fg) ≤ deg f + deg g; equality holds when R has no zero divisors, e.g. R is a field.)
f(x)g(x) = sum_{i=0}^m sum_{j=0}^n a_i b_j x^{i+j}.
Collecting like powers x^k (where k = i + j) gives a finite polynomial
f(x)g(x) = c_0 + c_1 x + ... + c_{m+n} x^{m+n},
with coefficients
c_k = sum_{i=0}^k a_i b_{k-i},
where terms with indices outside the ranges of a_i or b_j are taken as zero. Each c_k is a finite sum in R, so f(x)g(x) is again a polynomial in R[x].
Thus polynomials are closed under multiplication. (Also deg(fg) ≤ deg f + deg g; equality holds when R has no zero divisors, e.g. R is a field.)
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