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8.2 INTEGRAL ROOTS

Let the coefficients a_i of the polynomial equation

a₀xⁿ + a₁xⁿ^-1 + ... + a_n_-1x + a_n = 0

be integers. Then it can be shown that the only possibilities for integral roots are the integral divisors of the last coefficient a_n. For example, an integer that is a root of

x⁴ + x³ + x² + 3x - 6 = 0
would have to be one of the eight integral divisors

of -6. Trial of each of these eight integers, as in Example 2 in Section 8.1, would show that 1 and -2 are the only integral roots. The work can be reduced, when one root is found, by substituting the quotient polynomial for the original polynomial in further work. Thus

shows that x⁴ + x³ + x² + 3x - 6 = (x - 1)(x³ + 2x² + 3x + 6). Hence, 1 is a root and the other roots are the roots of the equation x³ + 2x² + 3x + 6 = 0.

COVER PAGE

TABLE OF CONTENTS

INDEX

PROBLEMS FOR THIS SECTION

page 65

Wednesday, June 3, 1998