MATH 9
ASSIGNMENT 2: ROOTS
SEPTEMBER 22, 2013
Roots
Definition. A number a ∈ R is called a root of polynomial f (x) if f (a) = 0.
The following theorem was proved in class.
Theorem 1. If a is a root of f (x), then f (x) is divisible by (x − a): f (x) = (x − a)g(x).
It can be generalized to several roots:
Theorem 2. If x1 , x2 , . . . , xk — distinct roots of polynomial f (x), then f (x) is divisible by (x − x1 )(x −
x2 ) . . . (x − xk ).
In particular, if x1 , x2 , . . . , xn are n distinct roots of the polynomial of degree n, then f (x) = c(x − x1 )(x −
x2 ) . . . (x − xn ) for some constant c.
Some corollaries of these results are given in homework below.
Multiple roots
Definition. A number a is called a multiple root of a polynomial f (x), with multiplicity m, if f (x) is
divisible by (x − a)m and not divisible by (x − a)m+1 .
Roots of multiplicity one are also called simple roots; of multiplicity two, double roots.
For example, polynomial (x − 1)2 (x − 5) has a simple root x = 5 and double root x = 1.
The following theorem generalizes results of the previous section:
Theorem 3. If a1 , . . . , ak are distinct roots of polynomial f (x), and m1 , . . . , mk are multiplicities of these
roots, then f (x) is divisible by the product (x − a1 )m1 . . . (x − ak )mk .
It is frequently convenient, when listing roots of a polynomial, to list double root twice, triple root three
times, etc, for example, listing the roots of polynomial (x − 1)2 (x − 5) as x1 = 1, x2 = 1, x3 = 5. This is
called “listing the roots with multiplicities”. Then the previous result can be rewritten as follows:
Theorem. If x1 , . . . , xn are roots of polynomial f (x), listed with multiplicities, then f (x) is divisible by
(x − x1 ) . . . (x − xn ).
Homework
1. Do the long division of the following polynomials:
(x4 − 2x2 + x) ÷ (x + 2)
(x6 − 1) ÷ (x2 + x + 1)
2. Prove Theorem 2. Hint: start with k = 2; write f (x) = (x − x1 )g(x) and then show that x2 is a root
of g(x).
3. (a) Show that a polynomial of degree n can not have more than n roots.
(b) Show that if f, g are two polynomials of degree n, and x1 , . . . , xn+1 — collection of diffferent
points, and f (x1 ) = g(x1 ), . . . , f (xn+1 ) = g(xn+1 ), then f (x) = g(x) (i.e., all coefficients of
f (x), g(x) are the same).
4. Let f (x) = xn + an−1 xn−1 + · · · + a0 be a polynomial with integer coefficients.
(a) Show that if a = pq is a rational root of this polynomial, then q = 1, i.e. a must be integer.
[Hint: which of the terms in q n−1 f (a) are integer?]
(b) Show that one can write f (x) = (x − a)g(x), where g(x) is a polynomial with integer coefficients.
(c) Show that a is a divisor of a0 .
5. Use the previous problem to find all rational roots of the polynomial x4 − 3x3 + 7x2 − 22x + 3
(remember that the roots can be negative).
6. Find roots with multiplicities of the following polynomials. Factor these polynomials if possible.
x3 − 3x2 + 4
x4 − 5x3 + 6x2
x3 + 4x2 − x − 10
7. (a) Show that x99 − 1 is divisible by x − 1, by x3 − 1, by x11 − 1.
(b) Can you find any factors of the number 17957 − 1?
*8. Is the polynomial 1 + x4 + x8 + · · · + x4k divisible by 1 + x2 + x4 + · · · + x2k ? [Hint: remember the
n+1
formula from last time, 1 + y + y 2 + · · · + y n = y y−1−1 .]