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2.2. Division of Polynomials 59

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Explain how to perform the algorithm, how to read off the quotient,

and remainder and why it works.

Make up a number of long division problems involving polynomials.

Solve them by the long division algorithm and by Horner’s Method

of Synthetic Division, and check the results.

Establish:

Division Theorem. Let f and g be two polynomials over a field F and

suppose that deg g 2 1. Then there are polynomials q (quotient) and

r (remainder) such that

f =gq+r and degr<degg.

The Division Theorem was formulated for polynomials over a field.

The case of dividing the polynomial 3t2 + t + 1 by 2t - 1 in Z[t] shows

that it does not always hold for polynomials over an integral domain.

Formulate and prove a modified version of the theorem in this case.

In the case of dividing a polynomial f(t) by the binomial (t - c), the

remainder can be given by a formula f(c) involving the polynomial f

and the coefficients of the divisor. Derive a formula for the remainder

when f(t) is divided by (t - a)(t - b).

Let F be a field and let F[x, y] denote the ring of polynomials in

the variable x and y with coefficients in F. Suppose f (x, y) belongs

to F[x, y]. Apply the Factor Theorem to the ring F[x] to show that

f (x, x) = 0 if and only if (x - y) is a factor of f (x, y). More generally,

show that y - g(x) divides f(x, y) if and only if f(x,g(x)) = 0, for

g(x) in F[x].

Consider the symmetric homogeneous polynomial

f (x, Y, 4 = x2$ + x3y2 + x2.z3 + x3,z2 + y2z3 + y3z2.

To find a representation of this polynomial in terms of the elementary

symmetric polynomials

Sl(X, y, z) = x + y + z

s2(x, y, z) = xy + yz + zx

33(x, Y, z) = XYZ,

proceed as follows: