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.

Now you No calculator settles this one, but the pieces of it are computable. Try one below, or type your own.

Keep your own working

A free account adds notes on every lesson, a record of what you have finished, your solved problems in one place, and a tutor you can ask about this page. The maths itself is open to everyone, signed in or not.

Symbols used here

Tap any symbol for the full definition, a picture, and what every letter in it means.

Questions people ask

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.

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

More in 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