Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

448 ALGEBRAIC SYSTEMS [APP. B

Corollary B.16 also tells us thatf(a)=0 if and only if the remainderr=r(t)≡0. Accordingly:

Corollary B.17 (Factor Theorem): The scalara∈Kis a root off(t)if and only ift−ais a factor off(t).

The next theorem (proved in Problem B.31) tells us the number of possible roots of a polynomial.

Theorem B.18: Supposef(t)is a polynomial over a fieldK, and deg(f)=n. Thenf(t)has at mostnroots.

The following theorem (proved in Problem B.32) is the main tool for finding rational roots of a polynomial
with integer coefficients.

Theorem B.19: Suppose a rational numberp/q(reduced to lowest terms) is a root of the polynomial

f(t)=antn+···+a 1 t+a 0

where all the coefficientsan,...,a 1 ,a 0 are integers. Thenpdivides the constant terma 0 andq

divides the leading coefficientan. In particular, ifc=p/qis an integer, thencdivides the

constant terma 0.

EXAMPLE B.16

(a) Supposef(t)=t^3 +t^2 − 8 t+4. Assumingf(t)has a rational root, find all the roots off(t).

Since the leading coefficient is 1, the rational roots off(t)must be integers from among±1,±2,±4.

Notef( 1 )=0 andf(− 1 )=0. By synthetic division, or dividing byt−2, we get

2

∣

∣

∣

∣

1 + 1 − 8 + 4

2 + 6 − 4

1 + 3 − 2 + 0

Thereforet=2 is a root andf(t)=(t− 2 )(t^2 + 3 t− 2 ). Using the quadratic formula fort^2 + 3 t− 2 =0,

we obtain the following three roots off(t):

t= 2 ,t=(− 3 +

√

17 )/ 2 ,t=(− 3 −

√

17 )/ 2

(b) Supposeh(t)=t^4 − 2 t^3 + 11 t−10. Find all the real roots ofh(t)assuming there are two integer roots.

The integer roots must be among±1,±2,±5,±10. By synthetic division (or dividing byt−1 and

thent+2) we get

1

∣

∣

∣

∣

1 − 2 + 0 + 11 − 10

1 − 1 − 1 + 10

− 2

∣

∣

∣

∣

1 − 1 − 1 + 10 + 0

− 2 + 6 − 10

1 − 3 + 5 + 0

Thust=1 andt=−2 are roots andh(t)=(t−1)(t+2)(t^2 − 3 t+5). The quadratic formula witht^2 − 3 t+ 5

tells us that there are no other real roots. That is,t=1 andt=−2 are the only real roots ofh(t).

K[t] as a PID and UFD

The following theorems (proved in Problems B.33 and B.34) apply.

Theorem B.20: The ringK[t]of polynomials over a fieldKis a principal ideal domain (PID). That is, ifJis an
ideal inK[t], then there exists a unique monic polynomial d which generatesJ, that is, every
polynomialfinJis a multiple ofd.