81. All Elementary Mathematics - Study Guide - Algebra - Division Of Polynomial By L division of polynomial by linear binomial. Linear binomial. Bezout stheorem. Linear binomial is a polynomial of the first degree http://www.bymath.com/studyguide/alg/alg6.html

Home Math symbols Jokes Consulting ... Site map Division of polynomial by linear binomial Linear binomial. Bezout's theorem. Linear binomial is a polynomial of the first degree: ax+ b. If to divide a polynomial, containing a letter x , by a linear binomial x – b , where b is a number ( positive or negative ), then a remainder will be a polynomial only of zero degree, i.e. some number N , which can be found without finding a quotient. Exactly, this number is equal to the value of the polynomial, received at x b. This property is proved by Bezout’s theorem: a polynomial a x m + a x m + a x m +  + a m is divided by x – b with a remainder N = a b m + a b m + a b m +  + a m The p r o o f . According to the definition of division (see above) we have: a x m + a x m + a x m +  + a m x – b Q + N where Q is some polynomial, N is some number. Substitute here x = b , then x– b Q will be missing and we receive: a b m + a b m + a b m +  + a m = N . The r e m a r k . It is possible, that N = . Then b is a root of the equation: a x m + a x m + a x m +  + a m The theorem has been proved.

82. Math Help - Algebra - Synthetic Division - Technical Tutoring So, if a division of a polynomial results in a nonzero remainder, substituting thex that makes the divisor zero results in the value of the polynomial equal http://www.hyper-ad.com/tutoring/math/algebra/Synthetic Division.html

Technical Tutoring Home Site Index Advanced Books Speed Arithmetic ... Harry Potter DVDs, Videos, Books, Audio CDs and Cassettes Synthetic Division Basic Process of Dividing Polynomials Limitations Remainders Examples ... Recommended Books Basic Process of Dividing Polynomials Elsewhere, we introduced the concept of synthetic division. Now we are going to amplify the concept and develop it more fully. The basic idea is to take a "big" (higher order) polynomial and divide it into a "small" (lower order) polynomial. If it divides evenly, we have in effect partially factored the polynomial. dividend , the smaller is called the divisor . First, set up the division: Looking at the first term in the dividend, we ask, "How many times does the first term of the divisor (x) need to be multiplied to get the first term of the dividend (x )? The answer is x . Write x on top of the bar above the x term, multiply both terms of the divisor by x , and write each result below the dividend term with the same power of x. Subtract: Notice that the x . For the next step, we bring down the next term (-7x

83. Math Help - Algebra - Factoring Large Polynomials - Technical Tutoring Try each one quickly via synthetic division. If one or more turns outto really be a root, factor the polynomial as much as possible. http://www.hyper-ad.com/tutoring/math/algebra/General Polynomials.html

Technical Tutoring Home Site Index Advanced Books Speed Arithmetic ... Harry Potter DVDs, Videos, Books, Audio CDs and Cassettes General Polynomials Terminology and Notation Factoring Large Polynomials Fundamental Theorem of Algebra Rational Zeros Theorem ... Recommended Books Terminology and Notation First, we present some notation and definitions. A general polynomial has the form This function is really a mathematical expression rather than an equation since the f(x) to the left of the equals sign is just a label or abbreviation for the long expression to the right of the first equals sign. The large symbol to the right of the second equals sign is called the sigma notation, and reads, "sum the product of the kth a and the kth power of x from k=1 up to k=n". This notation comes in handy when we are adding up a large number of terms that look alike. equation zeros of f(x) or roots of the equation f(x) = 0. The distinction between these terms is small (albeit precise) and the terms are often used interchangeably. Suppose we find the n numbers (read this last expression as "the set of all complex x which make f(x) = 0"; the first two expressions are two different ways of listing the individual x’s) that are all the possible roots of the equation. Then, we can express the polynomial in a much simpler form:

84. Karl's Calculus Tutor - Notes And Basic Algebra Concepts polynomial Long division. You can apply a procedure called polynomial long divisionin order to divide a polynomial of greater degree by one of lesser degree. http://www.karlscalculus.org/notes.html

Prependix C: Basic Algebra Concepts Note: This page of Karl's Calculus Tutor has recently been reorganized. If you are here for Math Notation on the Web, click here. If you are here for How to Send Math Notation by Email, click here If you are here for Why Bother to Learn Calculus, click here. If you are here for Study Tips, click here. Contents of this Page Stuff You Should Already Know Equations The Distrubutive Law Equation of a Straight Line ... The Unity of Calculus Stuff You Should Already Know You can't build a house from the roof down. In order to learn calculus, you have got to be able to do algebra. If you have no confidence in your algebra ability, perhaps you ought to see your advisor about putting off calculus for a semester in order to get some remediation in algebra. If you are not sure whether you need remediation or not, I recommend that you review the following material. If it all comes back to you, great. But if it brings back bad memories of never having understood it in the first place, consider your options carefully. You could be in over your head. If you need more extensive brush-up on algebra than is offered below, try clicking on

85. Synthetic Division denom; (where num and denom are polynomials in the same single variable) then denomis divided into num giving a polynomial part and remainder divided by denom http://www.mathwright.com/book_pgs/book635.html

Microworld: Synthetic Division : (All in One) Click the Hyperlink above to visit the Microworld. Author James White A(X) = Q(x)*B(x) + R(x) The polynomial Q(x) is the quotient of A(x) by B(x) , and R(x) is the remainder . If the degree of B(x) is larger than the degree of A(x) then Q(x) = 0, and R(x) = A(x). In this command line Microworld, you may experiment with this basic fact. There is a command called Synthetic num, denom and there are four programs called: Divide(num, denom) Pquotient (num, denom) Premainder (num, denom) GCD The difference between a command and a program is this. A command is followed by its arguments without parentheses, and the result is printed in the MathEdit object. It also stores the result in the variable ANSWER. A program is followed by its arguments within parentheses, and it returns its result as a value, so that the result can be an argument to another command or a program. If you type the command: Synthetic num, denom;

86. Dividing Polynomials Dividing polynomials. To divide a polynomial by a polynomail we use a long divisiontechnique similar to the long division technique used in arithmetic. http://home.sprynet.com/~smyrl/POLY6.HTM

Dividing Polynomials Quotient of a Monomial by a Monomial To divide a monomial by a monomial divide numerical coefficient by numerical coefficient, divide powers of same variable using third law of exponents. See second law of exponents Examples: Quotient of a Polynomial by a Monomial To divide a polynomial by a monomial divide each of the terms of the polynomial by a monomial. Example: Quotient of a Polynomial by a Polynomial To divide a polynomial by a polynomail we use a long division technique similar to the long division technique used in arithmetic. The long division technique can not currently be illustrated adequately using HTML (Hyper Text Markup Language). Consult any algebra text for a discussion of this technique. Remember in starting the long division process Write dividend and divisor in terms of descending powers of variable leaving space for any missing powers of the variable or writing in the missing powers with coefficient zero. (If there is more than one variable, arrange dividend and divisor in terms of descending powers of one of the variables. This is beyond the scope of Math 390 at MC.) Divide first term of divisor into first term of dividend (On subsequent iterations into first term of difference). Place this answer above long division symbol.

87. Division Of Polynomials division of polynomials. PROBLEM. Given the N+1 coefficients of apolynomial of degree N in A.0,A.1, ,AN, and M+1 coefficients of http://www.geocities.com/SiliconValley/Garage/3323/aat/a_dpol.html

Division of polynomials PROBLEM Given the N+1 coefficients of a polynomial of degree N in A.0,A.1,...,A.N , and M+1 coefficients of a polynomial of degree M in B.0,B.1,...,B.M , divide the polynomial A. by the polynomial B. giving a quotient polynomial in Q.1,Q.2,...,Q.NmM , where NmM=N-M and remainder polynomial whose coefficients are in R.1,R.2,...,R.Mm1 , where Mm1=M-1 IMPLEMENTATION Unit: internal subroutine Global variables: input arrays A. B. ; output arrays Q. R. Parameters: positive integers N M Result: quotient polynomial in Q.1,Q.2,...,Q.NmM , where NmM=N-M ; remainder polynomial in R.1,R.2,...,R.Mm1 , where Mm1=M-1 POLDIV: procedure expose A. B. R. Q. parse arg N, M Q. = do J = to N; R.J = A.J; end do K = N - M to by -1 MpK = M + K; Q.K = R.MpK / B.M do J = M + K - 1 to K by -1 JmK = J - K; R.J = R.J - Q.K * B.JmK end end return EXAMPLE The following program N = 4; M = 2 A.0 = 6; A.1 = -5; A.2 = 4; A.3 = -3; A.4 = 2 B.0 = 1; B.1 = -3; B.2 = 1 call POLDIV N, M Quo = "" do J = N - M to by -1; Quo = Quo Q.J"*X**"J; end Rem = "" do J = M - 1 to by -1; Rem = Rem R.J"*X**"J; end

88. Polynomials Worksheets FOIL (Binomial by Binomial) Multiplication (polynomial by polynomial) Dividinga Monomial by a Monomial Dividing a polynomial by a Monomial Mixing Addition http://www.edhelper.com/polynomials.htm

Return to edHelper.com Operations with Polynomials! Algebraic Polynomial Fractions. Introduction to Monomials, Binomials, and Trinomials. Adding Polynomials, Subtracting Polynomials, Multiplying Polynomials, Using FOIL, Dividing Polynomials, Mixed Addition, Subtraction, Division, and Multiplication Worksheets, Polynomial Functions, Factoring, Graphing, Real Zeros, ... Every time you click to create a worksheet a New worksheet is created! Read our Printing Worksheets Tip Section! Also Visit: Algebra Worksheets Simplifying Algebraic Fractions (Some Polynomials) Reducing Algebraic Fractions to Lowest Terms (Warm Up) Reducing Algebraic Fractions to Lowest Terms (A Little More Difficult) Reducing Algebraic Fractions to Lowest Terms (Different Variables) Multiplying Algebraic Fractions ... Final Review Polynomial Worksheets Classify as a monomial, binomial, or trinomial Adding Polynomials Subtracting Polynomials Multiplication (Monomial by Polynomial) ... Final Review of Polynomials Polynomial Functions Worksheets Dividing Polynomials: Polynomial by a Monomial Dividing Polynomials: Polynomial by a Binomial Dividing Polynomials: Polynomial by a Quadratic Dividing Polynomials: Mix ... Graph. State the local maxima and minima

89. All Elementary Mathematics - Study Guide - Algebra - Division Of Polynomials ... division of polynomials. division of polynomials (quotient, remainder).Long division. division of polynomials. What means to divide http://www.bymath.com/studyguide/alg/alg5.html

Home Math symbols Jokes Consulting ... Site map Division of polynomials Division of polynomials (quotient, remainder). Long division. Division of polynomials. What means to divide one polynomial  P  by another Q ? It means to find polynomials M ( quotient ) and N ( remainder ), satisfying the two requirements:           1).  An equality  MQ + N = P    takes place;           2).  A degree of polynomial  N   is less than a degree of polynomial Q Division of polynomials can be done by the following scheme ( long division 1) Divide the first term 16 a of the dividend by the first term  4 a of the divisor;  the result  4 a  is the first term of the quotient. 2) Multiply the received term 4 a  by the divisor  4 a – a + 2; write the result  16 a a a   under the dividend, one similar term under another. 3)  Subtract terms of the result from the corresponding terms of the dividend and move down the next by the order term 7 of the dividend; the remainder is 12 a a + 4) Divide the first term 12 a of this expression by the first term 4 a of the divisor;  the result 3 is the second term of the quotient.

90. ABSTRACT ALGEBRA ON LINE: Polynomials 4.2.1. Theorem. division Algorithm For any polynomials f(x) and g(x) in Fx,with g(x) 0, there exist unique polynomials q(x),r(x) Fx such that. http://www.math.niu.edu/~beachy/aaol/polynomials.html

POLYNOMIALS Excerpted from Beachy/Blair, Abstract Algebra 2nd Ed. Chapter 4 Roots; unique factorization Construction of extension fields Polynomials over Q ... About this document Roots; unique factorization 4.1.1. Definition. Let F be a set on which two binary operations are defined, called addition and multiplication, and denoted by + and respectively. Then F is called a field with respect to these operations if the following properties hold: (i) Closure: For all a,b F the sum a + b and the product a b are uniquely defined and belong to F. (ii) Associative laws: For all a,b,c F, a+(b+c) = (a+b)+c and a (b c) = (a b) c. (iii) Commutative laws: For all a,b F, a+b = b+a and a b = b a. (iv) Distributive laws: For all a,b,c F, a (b+c) = (a b) + (a c) and (a+b) c = (a c) + (b c). (v) Identity elements: The set F contains an additive identity element, denoted by 0, such that for all a F, a+0 = a and 0+a = a. The set F also contains a multiplicative identity element, denoted by 1 (and assumed to be different from 0) such that for all a F

91. Math 1010 On-line - Long Division And The Euclidean Algorithm division of polynomials. Polynomials can be divided mechanically by longdivision, much like numbers can be divided. Numbers represented http://www.math.utah.edu/online/1010/euclid/

Peter Alfeld Department of Mathematics College of Science University of Utah ... Mathematics 1010 online The Euclidean Algorithm Euclid of Alexandria lived during the third century BC. The Algorithm named after him let's you find the greatest common factor of two natural numbers or two polynomials Division of polynomials Polynomials can be divided mechanically by long division , much like numbers can be divided. Numbers represented in decimal form are sums of powers of 10. Polynomial expressions similarly are sums of powers of the variable (let's say ). There are two main differences: A coefficient in the decimal representation of a number must be one of the 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. On the other hand, a coefficient of a polynomial may be any real (or even complex ) number. Ten of a certain power of 10 can be traded for one higher power of 10, or vice versa. For example, ten 10s can be traded for one 100. By comparison, no trading is possible for polynomials. This actually simplifies the process of division.

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