In class, we performed a few examples of long division with polynomials,
After some discussion about the long division algorithm,
we concluded the following:
Theorem 1 (Theorem 5.16 in the textbook). Let D = Q or R.
Suppose a(x) and b(x) are polynomials in D[x], and assume b(x) ≠ 0. Then there
are unique polynomials
q(x), r(x) ∈ D[x] satisfying the following two conditions:
a(x) = b(x)q(x) + r(x)
deg(r(x)) < deg(b(x)).
Note that if b(x) is a factor of a (x), then r(x) = 0. In this case, deg(r(x)) =
-∞, which is indeed less than
deg(b(x)), as b(x) is nonzero by assumption.
Proof. We must prove existence and uniqueness of the polynomials q(x) and r(x).
As we discussed in class, the long division algorithm yields quotient polynomial
q(x) and a remainder
polynomial r(x) satisfying the conditions of the theorem, giving existence.
The crux of the proof of uniqueness is the following observation:
Lemma. Let . If
is nonzero and
then is the zero polynomial.
Proof of Lemma. We show the contrapositive: if is nonzero, then
Indeed, if is nonzero then
natural numbers, so .
To show uniqueness, suppose that q(x), q'(x), r(x), r'(x) satisfy
a(x) = b(x)q(x) + r(x), deg(r(x)) < deg(b(x))
a(x) = b(x)q'(x) + r'(x), deg(r'(x)) < deg(b(x)).
It follows that
b(x)(q(x) - q'(x)) = r'(x) - r(x).
Observe that the degree of r'(x) - r(x) at most the largest of deg(r'(x)) and deg(r(x)). (It may not be exactly
deg(r(x)) or deg(r'(x)) because of cancellation.) Both deg(r'(x)) and deg(r(x))
are less than deg(b(x)),
so deg(r'(x) - r(x)) < deg(b(x)). Hence
(Recall that s(x) is a factor of p (x) in D[x] if there is another polynomial
t(x) in D[x] so p(x) = s(x)t(x).)
Example. The polynomials x2 - 2, x, 2x, 4x, x2 + x - 1, and x2 + x + 1 are all
irreducible in Z[x].
The polynomials x2 - 2 and x2 + x - 1 are irreducible in Q[x] and Z[x], but
are reducible in R[x].
The polynomial x2 + x + 1 is irreducible in Z[x], Q[x], and R[x].
Polynomials, like the integers , are closed under addition, subtraction , and
multiplication; and are not closed
under division. However, there are certain numbers that we can always divide by
and still receive a polynomial;
these numbers depend on the coefficient domain.
Definition 3. An element k ∈ D is a unit if for every polynomial p(x) ∈ D[x],
we have p(x) ∈ D[x].
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