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5. Polynomials
Our aim in this section is to reveal that there are close similarities between the ring Z of integers and the ring R[x] of polynomials with coefficients in R.
5.1 Polynomials.
Examples: The following are (real) polynomials:
x^2 , 2 + πx + 7x^2 , √ 2 + 3x − 5 x^3 =
2 + 3x + 0.x^2 − 5 x^3 , 42
but
1 + x + x^2 +... + xr^ +... =
∑^ ∞
r=
xr
(which has infinitely many terms) is NOT a polynomial.
Definition: Formally, a (real) polynomial in the variable x is an expression
f := a 0 + a 1 x + · · · + anxn,
where n is a non-negative integer and ai ∈ R for i = 0,... , n. We may alternatively write polynomials ‘top-down’: f = anxn^ + an− 1 xn−^1 + · · · + a 1 x + a 0.
Polynomials of the form a 0 , where a 0 ∈ R, are called constants; the constant 0 is the zero polyno- mial. The set of all polynomials in the variable x and coefficients in R is denoted R[x].
A real polynomial a 0 + a 1 x + · · · + anxn^ gives rise to a polynomial function from R to R:
y 7 −→ a 0 + a 1 y + · · · + anyn^ (y ∈ R).
Given any a ∈ R, we have a map εa : R[x] → R (evaluation at a): εa(f ) := f (a). From now on we shall not distinguish between a polynomial (formally an abstract string of symbols) and the polynomial function it determines. At Mods level doing this causes no problems, but anyone who is concerned that this is a bit slapdash may like to read the account of polynomials in P.J. Cameron, Introduction to Algebra.
Arithmetic operations on polynomials. Binary operations of addition and multiplication are defined on R[x] in the following way. Let
f = a 0 + a 1 x + · · · + anxn^ and g = b 0 + b 1 x + · · · + bmxm
be two polynomials.
- Addition: let k = max{n, m} and define ai = 0 for any i such that n < i 6 k and bi = 0 for any i such that m < i 6 k. Then the sum f + g is defined to be the polynomial c 0 + c 1 x + · · · + ckxk^ where ci = ai + bi for i = 0,... , k = max{n, m}.
- Multiplication: the product f · g is defined to be the polynomial c 0 + c 1 x + · · · + ckxk^ where cj =
∑j i=0 aibj−i^ for all^ j^ = 0,... , k^ =^ n^ +^ m. These are exactly the rules you use to do addition and multiplication of polynomial functions and, with respect to these rules, the polynomials can be viewed as a subring of the ring RR^ of all real- valued functions f : R → R, with pointwise addition and multiplication (recall the example in 2.6).
Thus R[x] is a commutative ring with identity, the zero and identity being the zero polynomial and the constant polynomial 1. Note that the definition of multiplication and the fact that R has no zero divisors together imply that R[x] has no zero divisors.
The degree of a (non-zero) polynomial. A non-zero polynomial f can be written as f = a 0 + a 1 x + · · · + anxn, where an 6 = 0, and we call n the degree of f and denote it by deg f (∂f in some books). Note that the degree of the zero polynomial is not defined. For non-zero f, g ∈ R[x],
deg(f + g) 6 max{deg f, deg g} if f + g 6 = 0; deg(f · g) = deg f + deg g.
We collect together what we have already, and a bit more, into a theorem. Compare this with Theorem 2.10. To prove (ii), exploit the fact that deg(f · g) = deg f + deg g for non-zero polynomials f and g.
5.2 Theorem (R[x] as a ring).
(i) R[x] is a commutative ring with 1 , with multiplicative identity the constant polynomial 1. (ii) f ∈ R[x] has a multiplicative inverse (that is, f is a unit) if and only if f is a non-zero constant. (iii) R[x] is not a field but is an integral domain, that is, (i) holds and
(ID) f · g = 0 implies f = 0 or g = 0.
Remark: All the rules for addition and multiplication, and the theorem above, would work equally well if the coefficients in our polynomials were drawn from Q or C, or any other field, and the above theorem would work too.
f 1 = q 1 g + r 1 with r 1 = 0 or deg r 1 < deg g.
We deduce that f 1 −
an bm
xn−m^ · g = q 1 g + r 1 ,
and hence
f =
an bm
xn−m^ + q 1
g + r 1.
We can then take q = (an/bm)xn−m^ + q 1 and r = r 1 ; this satisfies r = 0 or deg r < deg g.
Flow chart for the proof of existence of q and r
INPUT: polynomial f
@R
f = 0 q = r = 0 HOME
f 6 = 0
inductive hypothesis result true for polys of deg < n
@R
deg f < deg g q = 0, r = f HOME
deg f > deg g
f 1 := f − a bmn xn−mg
@R
f 1 = 0 r = 0, q = a
n bm x
n−m
HOME
f 1 6 = 0
by ind. hyp.: f 1 = q 1 g + r 1 r = r 1 , q = a bmn xn−m^ + q 1 HOME
Uniqueness of q and r:
This proof parallels that given for Z in 4.6. Assume there are two pairs of polynomials q, r and ˜q, ˜r such that f = q · g + r where either r = 0 or deg r < deg g, and f = ˜q · g + ˜r where either ˜r = 0 or deg ˜r < deg g. Hence we get (q − q˜)g = ˜r − r. Then, either q = ˜q and so ˜r = r; or deg(q − q˜) > 0 and so deg[(q − q˜)g] > deg g by the facts about degrees (see 5.1), while ˜r − r is either 0 or has degree not greater than deg g − 1, which cannot occur.
We now give an important application of the Division Algorithm.
5.5 The Remainder Theorem. Let f ∈ R[x] and let a ∈ R. Then there exists q ∈ R[x] such that f = q · (x − a) + f (a).
In particular, (x − a) is a factor of f , in the sense that f = (x − a)g for some g ∈ R[x], if and only if f (a) = 0.
Proof. The division algorithm for polynomials tells us that we can find polynomials q and r such that f = q · (x − a) + r with either r = 0 or deg r < deg(x − a) = 1. Hence either r = 0 or r is a non-zero constant. Putting x = a we get r = 0 if and only if f (a) = 0.
As usual, we say that a ∈ R is a root of f if f (a) = 0.
. Corollary to Remainder Theorem. Let f ∈ R[x] be of degree n. Then f has at most n real roots, counted according to multiplicity.
Proof. Argue by induction on the degree of f.
Remark: The Fundamental Theorem of Algebra. As you probably know, every polynomial f of degree n having real (or complex) coefficients has n roots, not necessarily distinct, in C and so can be written as a product of n linear factors:
f = c(x − α 1 )... (x − αn) (for some α 1 ,... , αn, c ∈ C, c 6 = 0).
This major theorem can be proved, easily, using deep results on analysis in the complex plane (Part A Analysis, next year).
5.6 Divisors of polynomials. For definiteness we work with R[x], but we could replace R with any field K, for example K = Q or C.
Definition: A non-zero polynomial g divides (or is a factor of) the polynomial f if there exists a polynomial q such that f = qg, that is, if r = 0 in the division algorithm. Notation: g|f.
Divisors Lemma for polynomials. Let f , g and h be non-zero polynomials in R[x].
(i) f | 1 if and only if f is a non-zero constant. (ii) Assume f |g and g|f. Then there exists α ∈ R r { 0 } such that f = αg. (iii) Assume h|f and h|g, then h|(ma + nb) for all m, n ∈ R[x].
Proof. The proof parallels that of the Divisors Lemma for integers, drawing on Theorem 5.2(ii) and (iii) and the division algorithm for polynomials. The details are left as an exercise.
5.8 Highest common factor theorem for polynomials.
(i) Let f and g be non-zero polynomials in R[x]. Then a highest common factor, hcf(f, g), of f and g exists; (ii) (the hcf formula) there exist polynomials m and n such that
hcf(f, g) = mf + ng.
Proof. Without loss of generality we may assume that deg g 6 deg f. In what follows, all quantities appearing are polynomials and at each step the division algorithm is invoked. We assume that rk is the first remainder to be zero and stop as soon as such a zero remainder occurs, so that deg r 0 ,... , deg rk− 1 are defined. We can write
f = q 0 g + r 0 where deg r 0 < deg g g = q 1 r 0 + r 1 where deg r 1 < deg r 0 r 0 = q 2 r 1 + r 2 where deg r 2 < deg r 1
......... rk− 3 = qk− 1 rk− 2 + rk− 1 where deg rk− 1 < deg rk− 2 rk− 2 = qkrk− 1 + rk where 0 = rk
The Going Down Lemma from 3.1, applied to the degrees deg r 0 , deg r 1 ,... , ensures that the process does indeed terminate in a finite number of steps. The Invariance Lemma tells us that
hcf(f, g) = hcf(g, r 0 ) =... = hcf(rk− 2 , rk− 1 ) = rk− 1.
To obtain the hcf formula we retrace our steps: write rk− 1 in terms of rk− 2 , then substitute the resulting formula for rk− 1 into the equation for rk− 3 and so on until a formula in f and g of the required form is obtained.
5.9 Irreducible polynomials. A non-zero polynomial p ∈ R[x] of degree k is said to be irreducible if its only divisors are polynomials of degree 0 (ie non-zero constants) and degree k. Irreducible polynomials play the role for R[x] that primes do for Z and there is a unique factorisation theorem for R[x] (or K[x], where K is any field). This is a parallel to the Fundamental Theorem of Arithmetic for Z. We shan’t pursue existence and uniqueness of factorisation into irreducibles any further in this course.
