1 - x + 6x3
-7
x
are all examples of polynomials. Here's a precise definition.

A polynomial in an indeterminate (or variable x) with coefficients in a ring R is a formal expression

f=a0 + a1x + a2x2 + ... + anxn
where n is a nonegative integer, and each ai is in R. The constant term of f is a0 and ai is called the coefficient of xi in f. The coefficients of xN in f, for N>n are defined to be zero. Given a second formal expression
g=b0 + b1x + b2x2 + ... + bmxn
Then we say that f=g if and only if for each i the coefficents of xi in f and g are equal. (This is what we mean when we speak of equating coefficients in a polynomial equation.)

The polynomial f is called zero if all its coefficients are zero. A nonzero polynomial has a degree which is defined to be the largest nonnegative integer t such that the coefficient of xt in f is nonzero. By convention the zero polynomial has degree -infinity. For short we write deg f for the degree of f.

We define rules to add and multiply the polynomials f,g as follows

f + g = (a0+b0) + (a1+b1)x + ... + (an+bn)xn + ... +bmxm
if m>=n and
f . g = c0 + c1x + c2x2 + ... + cn+mxn+m
where
ck=a0bk + a1bk-1 +...+ arbk-r +...+ a0bk

These rules makes the collection of all such polynomials into a ring. This ring is denoted by R[x] and is called the polynomial ring in x with coefficients in R. Note that the collection of constant terms is a subring of R[x] which is isomorphic to R. We always identify R with this subring and think of R as sitting inside R[x].

Note that deg(f+g) <= max{ deg f, deg g }. If R is an integral domain then deg(fg) = deg f + deg g.

Polynomials in several variables

Starting with R[x] we could form the polynomial ring R[x][y] in a new indeterminate, y, say. The elements of this ring have a unique expression as a sums of terms

ai,jxiyj
with (i,j) ranging over finitely many pairs of nonnegative inetgers and ai,j nonzero in R.

Clearly, if we had started with R[y] and then formed R[y][x] we would have obtained the same ring. This ring is denoted by R[x,y]

More generally, we can inductively define

R[x1,...,xn]=R[x1,...,xn-1][xn]
the polynomial ring in several indeterminates. A typical element of this ring can be written uniquely as a sum of monomials
ai1,i2,...,inx1i1...xnin
where (i1,...,in) range over finitely many tuples of integers and ai1,i2,...,in is nonzero in R.

The total degree of such a monomial is defined to be i1+...+in and the total degree of a nonzero polynomial is the largest total degree of a monomial.

anxn + an-1xn-1 + an-2xn-2. . . a1x + a0

The degree of a polynomial is the highest total of powers of variables (x, y, etc.) of a single term, so in the polynomial 2xy2 + x2 the degree is three (in the first term, x has a power of one). The standard form of a polynomial is when you write it with the degrees descending (x2 + x + 3, not x + x2 + 3)

To factor a polynomial (If you already know how to then skip down to the AC method. You'll like it. A lot.) you first factor out the common factor, if there is one, using the distributive property:

Ex 1) 2x2 + 4x = 2x(x + 2)
Ex 2) 2x2 + 6x + 8 = 2(x2 + 3x + 4)

With a binomial (two terms, as in Ex 1) that's all. If you have a trinomial (three terms, as in Ex 2) you're just getting started.

You usually have to find two binomials (B1 and B2) whose first terms multiply to the first term of your trinomial, last terms multiply to the last term of the trinomial, and B1's first term times B2's last term plus vice versa equals the middle term (FOIL users: Inside + Outside=Middle)

Ex 3) x2 + 3x + 2 = (x + 1)(x + 2)

If the first term of your trinomial has a coefficient (a) of 1--as shown above--then the first terms of the binomials are x. Otherwise, you have to play around searching for the proper factors to get it right. That's where the following method comes in:

The AC Method

First factor out the common factor. Always, always, always do this.
Now you have ax2 + bx + c a isn't 1.
Change it to x2 + bx + ac. (If you're stuck wondering how the hell to move the a all the way over to the c, don't bother. Just do it.)
Factor x2 + bx + ac into your (presumably) two binomials. Then stick a back into the first terms of both of them, factor out the common factor and toss it out. You're done.

Ex 4) 6x2 + 2x-4
2(3x2 + x-2) (Factor out common factor)
2(x2 + x-6) (move a to third term)
2(x + 3)(x-2) (factor)
2(3x + 3)(3x-2) (put a back into first terms)
2(x + 1)(3x-2) (factor out and delete common factor)
If you're planning on using the AC Method a lot you may want to work on your factoring large numbers because ac is often rather large.

Now, I know you're thinking, "What if I have a four-term (or more) polynomial?" Easy: Take a few terms, and slap parenthesis around them (Hint, put together terms that have common factors or that look like they'll factor easily.)

Ex 5) 2x3 - 3x2 + 4x - 6
(2x3 - 3x2) + (4x - 6)
x2(2x - 3) + 2(2x - 3)
(x2 + 2)(2x - 3)

That last example (first and last steps anyway) was taken from College Algebra by Michael Sullivan because I was having a heck of a time making up a good example. (I'm always coming up with prime polynomials in my example and having to modify them so I can factor them. I wish my math teacher had let me do that in my homework.)

Now you need to do some heavy memorising. These are special polynomials and how to factor them. Knowing how to recognise them will help you enormously, both in multiplication and factoring:

Difference of Squares: x2 - a2 = (x - a)(x + a) (Ex 6) x2 - 144 = (x + 12)(x - 12))
Perfect Squares: x2 ± 2ax + a2 = (x ± a)2
Unnamed, but bears remembering: x2 (a + b)x + ab = (x + a)(x + b)
Unnamed, but bears remembering: acx2 + (ad + bc)x + bd = (ax + b)(cx + d)
Perfect Cubes: x3 + 3ax2 + 3a2x + a3 = (a + x)3, x3 - 3ax2 + 3a2x - a3 = (a - x)3
Sum of Two Cubes: x3 + a3 = (x + a)(x2 - ax + a2)
Difference of Two Cubes: x3 - a3 = (x - a)(x2 + ax +a2)

Take the coefficients of (x + y)n and look at the nth row of Pascal's Triangle (the "1" at the top is 0th). Cute and useful. (m_turner is smart.)


Please /msg with additions and/or corrections.
This writeup is licensed under the GNU Free Documentation License.

Pol`y*no"mi*al (?), n. [Poly- + -nomial, as in monomial, binomial: cf. F. polynome.] Alg.

An expression composed of two or more terms, connected by the signs plus or minus; as, a2 - 2ab + b2.

 

© Webster 1913.


Pol`y*no"mi*al, a.

1.

Containing many names or terms; multinominal; as, the polynomial theorem.

2.

Consisting of two or more words; having names consisting of two or more words; as, a polynomial name; polynomial nomenclature.

 

© Webster 1913.

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