Your Algebra Homework Can Now Be Easier Than Ever!

Polynomials

1 Definition of Polynomial

Let D denote the number system we are working over. For our purposes, it will usually denote Z, Q, or R.
Definition 1. A polynomial in x over D is a finite sum



where kj ∈ D for all 0 ≤ j ≤ n and n is a natural number; we use D[x] to denote the collection of all polynomials in
x over D.

The system D is called the coefficient domain .

The degree of each nonzero term is the exponent of x ; the degree of a nonzero polynomial is the highest degree of its
terms.

The leading coefficient of a polynomial is the coefficient associated to its highest degree term.

The degree of a polynomial p(x) is sometimes denoted as deg(p(x)).

Examples of polynomials in x over Q and their degrees are:

When two nonzero polynomials p(x) and q(x) are multiplied, the degree of the resulting polynomial is the
sum of the original polynomials:

deg(p(x)q(x)) = deg(p(x)) + deg(q(x)). (* )

For this reason, the degree of the zero polynomial is often taken to be either undefined or -∞: otherwise,
products with the zero polynomial would violate rule (*). For the purposes of this class, we will define the
degree of the zero polynomial to be -∞, as it will simplify the statements of some theorems.

2 The Division Algorithm for Polynomials


In class, we performed a few examples of long division with polynomials, obtaining:

After some discussion about the long division algorithm, we concluded the following:

Theorem 1 (Theorem 5.16 in the textbook). Let D = Q or R.

Suppose a(x) and b(x) are polynomials in D[x], and assume b(x) ≠ 0. Then there are unique polynomials
q(x), r(x) ∈ D[x] satisfying the following two conditions:

a(x) = b(x)q(x) + r(x)

deg(r(x)) < deg(b(x)).

Note that if b(x) is a factor of a (x), then r(x) = 0. In this case, deg(r(x)) = -∞, which is indeed less than
deg(b(x)), as b(x) is nonzero by assumption.

Proof. We must prove existence and uniqueness of the polynomials q(x) and r(x).

As we discussed in class, the long division algorithm yields quotient polynomial q(x) and a remainder
polynomial r(x) satisfying the conditions of the theorem, giving existence.

The crux of the proof of uniqueness is the following observation:

Lemma. Let . If is nonzero and



then is the zero polynomial.

Proof of Lemma. We show the contrapositive: if is nonzero, then



Indeed, if is nonzero then are
natural numbers, so .

To show uniqueness, suppose that q(x), q'(x), r(x), r'(x) satisfy

a(x) = b(x)q(x) + r(x), deg(r(x)) < deg(b(x))

a(x) = b(x)q'(x) + r'(x), deg(r'(x)) < deg(b(x)).

It follows that

b(x)(q(x) - q'(x)) = r'(x) - r(x).

Observe that the degree of r'(x) - r(x) at most the largest of deg(r'(x)) and deg(r(x)). (It may not be exactly
deg(r(x)) or deg(r'(x)) because of cancellation.) Both deg(r'(x)) and deg(r(x)) are less than deg(b(x)),
so deg(r'(x) - r(x)) < deg(b(x)). Hence

deg(b(x)(q(x) - q'(x))) = deg(r'(x) - r(x)) < deg(b(x)).

We deduce that q(x) - q'(x) = 0 using our lemma. It follows that r'(x) - r(x) = 0. We have shown
uniqueness.

Remark. The above theorem does not hold for Z[x]. For example, let a(x) = x2 and b(x) = 2x.

3 Irreducible Polynomials, Factorizations, and Units

3.1 Irreducible Polynomials


“Prime” is to “integer” as “irreducible” is to “polynomial”.

Definition 2. A nonconstant polynomial p(x) ∈ D[x] is irreducible if it has no nonconstant factors in D(x) other
than multiples of itself .

If a polynomial is not irreducible, it is reducible .

(Recall that s(x) is a factor of p (x) in D[x] if there is another polynomial t(x) in D[x] so p(x) = s(x)t(x).)

Example. The polynomials x2 - 2, x, 2x, 4x, x2 + x - 1, and x2 + x + 1 are all irreducible in Z[x].
The polynomials x2 - 2 and x2 + x - 1 are irreducible in Q[x] and Z[x], but are reducible in R[x].
The polynomial x2 + x + 1 is irreducible in Z[x], Q[x], and R[x].

3.2 Units

Polynomials, like the integers , are closed under addition, subtraction , and multiplication; and are not closed
under division. However, there are certain numbers that we can always divide by and still receive a polynomial;
these numbers depend on the coefficient domain.

Definition 3. An element k ∈ D is a unit if for every polynomial p(x) ∈ D[x], we have p(x) ∈ D[x].

The units of Z[x], Q[x], and R[x] are:

polynomials units
Z[x] ±1
Q[x] elements of Q
R[x] elements of R

Analogously, the units of Z are ±1.

Prev Next

Start solving your Algebra Problems in next 5 minutes!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of September 17th you will pay only $39.99 instead of our regular price of $74.99 -- this is $35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:



Algebra Helper can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath
Check out our demo!
 
"It really helped me with my homework.  I was stuck on some problems and your software walked me step by step through the process..."
C. Sievert, KY
 
 
Sofmath
19179 Blanco #105-234
San Antonio, TX 78258
Phone: (512) 788-5675
Fax: (512) 519-1805
 

Home   : :   Features   : :   Demo   : :   FAQ   : :   Order

Copyright © 2004-2021, Algebra-Answer.Com.  All rights reserved.