maths.free › Abstract Algebra › Polynomials › The Division Algorithm

The Division Algorithm

Recall that the division algorithm for integers () says that if a and b are integers with b \gt 0, then there exist unique integers q and r such that a = bq + r, where 0 \leq r \lt b.

The Division Algorithm

Recall that the division algorithm for integers () says that if \(a\) and \(b\) are integers with \(b \gt 0\), then there exist unique integers \(q\) and \(r\) such that \(a = bq + r\), where \(0 \leq r \lt b\). The algorithm by which \(q\) and \(r\) are found is just long division. A similar theorem exists for polynomials. The division algorithm for polynomials has several important consequences. Since its proof is very similar to the corresponding proof for integers, it is worthwhile to review at this point.

Example

The division algorithm merely formalizes long division of polynomials, a task we have been familiar with since high school. For example, suppose that we divide \(x^3 - x^2 + 2 x - 3\) by \(x - 2\).

			\(x^2\)	\(+\)	\(x\)	\(+\)	\(4\)
\(x\)	\(-\)	\(2\)	\(x^3\)	\(-\)	\(x^2\)	\(+\)	\(2x\)	\(-\)	\(3\)
			\(x^3\)	\(-\)	\(2x^2\)
					\(x^2\)	\(+\)	\(2x\)	\(-\)	\(3\)
					\(x^2\)	\(-\)	\(2x\)
							\(4x\)	\(-\)	\(3\)
							\(4x\)	\(-\)	\(8\)
									\(5\)

Hence, \(x^3 - x^2 + 2 x - 3 = (x - 2) (x^2 + x + 4 ) + 5\).

Let \(p(x)\) be a polynomial in \(F[x]\) and \(\alpha \in F\). We say that \(\alpha\) is a zero or root of \(p(x)\) if \(p(x)\) is in the kernel of the evaluation homomorphism \(\phi_{\alpha}\). All we are really saying here is that \(\alpha\) is a zero of \(p(x)\) if \(p(\alpha) = 0\).

Corollary

Let \(F\) be a field. An element \(\alpha \in F\) is a zero of \(p(x) \in F[x]\) if and only if \(x - \alpha\) is a factor of \(p(x)\) in \(F[x]\).

Proof

Suppose that \(\alpha \in F\) and \(p( \alpha ) = 0\). By the division algorithm, there exist polynomials \(q(x)\) and \(r(x)\) such that \[\begin{aligned}\end{aligned}\] and the degree of \(r(x)\) must be less than the degree of \(x -\alpha\). Since the degree of \(r(x)\) is less than \(1\), \(r(x) = a\) for \(a \in F\); therefore, \[\begin{aligned}\end{aligned}\]. But \[\begin{aligned}\end{aligned}\] consequently, \(p(x) = (x - \alpha) q(x)\), and \(x - \alpha\) is a factor of \(p(x)\).

Conversely, suppose that \(x - \alpha\) is a factor of \(p(x)\); say \(p(x) = (x - \alpha) q(x)\). Then \(p( \alpha ) = 0 \cdot q(\alpha) = 0\).

Notice the similarity between the proof of and the proof of .

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.

Tsopano inu Sikuti pali calculator yomwe imatha kufotokoza izi, koma zigawo zake zimatha kuwerengedwa. Timafuna kuyesera imodzi pansipa, kapena tidzalemba ya ife.

Kusunga ntchito yanu yokha

M’malo mwake, maakaunti aulere amawonjezera malemba pazophunzira zonse, mnda wa zomwe mwamaliza, mavuto anu omaliza m’malo limodzi, ndi m’bale amene mungafunse za nkhaniyo. Maphunziro a matekinoloje ndi otsegulira kwa aliyense, olembetsa kapena osalembetsa.

Kulembetsa Kulowa

Zithunzi

Pitani pa dzina lililonse lachifaniziro kuti mudziwe tanthauzo lake, chithunzi chake, ndi zimene limatanthauza.

Mafunso omwe anthu amafunsa

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Zigawo za m'nkhaniyi ndi zochokera Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Kuphatikizapo ndi kufotokozanso pano; zolakwika ndi zathu.

Zambiri pa Abstract Algebra

Groups Subgroups, cosets and Lagrange's theorem Cyclic groups and permutation groups Homomorphisms, normal subgroups and quotient groups Rings and fields Galois theory: why the quintic has no formula