Polynomials and fields · Definition
Polynomial division algorithm
Before you start you need to know
It is the polynomial version of integer division with remainder, and the tool that gives \(F[x]\) an arithmetic as convenient as that of \(\mathbb{Z}\).
Definition
Given polynomials \(f\) and \(g\neq 0\) over a field \(F\), there exist a unique quotient \(q\) and a unique remainder \(r\) satisfying the equality below, with \(r\) zero or of degree less than that of \(g\).
The procedure is the one from school: divide the leading term of \(f\) by that of \(g\), subtract that multiple of \(g\), and repeat with what is left until the degree of the remainder drops below that of \(g\). Each step requires dividing leading coefficients, which is why a field is assumed. For example, \(x^3-2x^2+x-5=(x-2)(x^2+1)-3\): the quotient is \(x^2+1\) and the remainder is \(-3\).
Uniqueness comes from comparing degrees: if there were two divisions, their difference would give a nonzero multiple of \(g\) of degree less than \(\deg g\), which is absurd. The same argument works over any commutative ring as long as the leading coefficient of \(g\) is a unit; without that it can fail, as when dividing \(x^2\) by \(2x\) in \(\mathbb{Z}[x]\).
A very common special case is division by \(x-a\): the remainder is the constant \(f(a)\), which gives the remainder theorem (\(f(a)=0\) if and only if \(x-a\) divides \(f\)) and the bound that a polynomial of degree \(n\) has at most \(n\) roots in a field. The algorithm also computes greatest common divisors in \(F[x]\) by repeated division, and it is the model for the next concept: the Euclidean domain.
Think about it
If \(f\) has degree 5 and \(g\) has degree 2, what is the degree of the quotient?
Degree 3: the leading term of \(f\) is that of \(gq\), so \(\deg q=\deg f-\deg g\).
Why does \(x-a\) divide \(f\) exactly when \(f(a)=0\)?
Dividing gives \(f=(x-a)q+c\) with \(c\) constant; evaluating at \(a\) yields \(c=f(a)\), which is zero if and only if the remainder is zero.
Exercises 10
- When dividing \(x^{3} - 2 x^{2} + x - 5\) by \(x-2\) in \(\mathbb{Q}[x]\), what is the remainder?
- Compute the quotient of dividing \(2 x^{3} + 3 x^{2} - x + 4\) by \(x+1\) in \(\mathbb{Q}[x]\).
- When dividing \(f\) by a polynomial \(g\) of degree 3 over a field, what is known about the remainder \(r\)?
- Over \(\mathbb{Q}\), what is the division of \(x^{4} + 1\) by \(x^{2} + x + 1\)?
- In \(\mathbb{Z}_5[x]\), what are the quotient \(q\) and remainder \(r\) of dividing \(x^{3} + 2 x + 1\) by \(2 x + 3\)?
- What is the remainder of dividing \(x^{101}+x^2+1\) by \(x+1\) in \(\mathbb{Q}[x]\)?
- In \(\mathbb{Z}[x]\), we want to divide \(x^2\) by \(2x\) with \(r=0\) or \(\deg r<1\), and \(q,r\in\mathbb{Z}[x]\). What happens?
- Compute the remainder of dividing \(x^{5} + 3 x^{3} - x + 2\) by \(x^{2} + 1\) in \(\mathbb{Q}[x]\).
- In the uniqueness proof, if \(f=gq_1+r_1=gq_2+r_2\) with \(q_1\ne q_2\), what contradiction appears?
- In \(\mathbb{Z}_3[x]\), how many different remainders are possible when dividing by a polynomial of degree 2?
What comes next
References
- David S. Dummit, Richard M. Foote, Abstract Algebra, 3rd edition, Wiley, 2004. §9.2
- Joseph A. Gallian, Contemporary Abstract Algebra, 9th edition, Cengage, 2017. cap. 16
- I. N. Herstein, Abstract Algebra, 3rd edition, Wiley, 1996. §3.9
- John B. Fraleigh, A First Course in Abstract Algebra, 7th edition, Pearson, 2002. §23
- Serge Lang, Algebra, Revised 3rd edition, Springer, 2002. cap. IV §1