I don't doubt that it does, I'm just curious where. Polynomial long division is used in error-control coding theory and practice. Perhaps the most common use is for error detection via cyclic redundancy check CRC methods. The receiver checks whether the received sequence of bits is in fact a polynomial divisible by the CRC polynomial by carrying out a polynomial long division. If the remainder is zero, the data portion is accepted.
If the remainder is nonzero, the data is discarded and a re-transmission of the packet is requested. For completeness I will say that at the transmitter, the CRC check sum bits are also found by doing a polynomial long division.
More details can be found here. If people have ideas as to how these ubiquitous calculations can be speeded up, many engineers will be happy to hear from you. The theory of polynomial long division is very relevant in the construction of algebraic extensions of fields, though you may not need to actually long divide any polynomials, just understand how it works. One case where polynomial long division shows up is in the study of Transfer Functions.
Additionally, polynomial long division is encountered in computational algebraic geometry, and in particular in the context of Hilbert series computation. In my case I think it mainly came up when integrating rational functions, which was a core topic in 2nd semester calculus. To a lesser extent, integrating rational functions also came up in courses such as elementary ordinary differential equations, engineering separation of variables PDE courses, "advanced mathematics for engineering" courses, etc.
I believe I may have divided polynomials in a complex variables course as well. However, by far the most use I made of dividing polynomials was when helping friends with lower level math for free , tutoring students in lower level math for pay , and teaching lower level math classes.
Indeed, with the exception of one year in the late s when I taught arithmetic and geometry in a high school , I think I may have taught long division or at least strongly reviewed it for students in at least one class each semester for over 20 years of teaching. It's time it was kicked out of the school math curriculum. Here is an example:. Just what is needed to match the original numerator. Sign up to join this community.
The best answers are voted up and rise to the top. The best way to do this is to explicitly work out the equation. Synthetic division is a technique for dividing a polynomial and finding the quotient and remainder. To use the remainder theorem, one must first perform division, which is a bit of work. A shorthand way to perform long division is synthetic division. It uses less writing and fewer calculations.
It also takes significantly less space than long division. Most importantly, the subtractions in long division are converted to additions by switching the signs at the very beginning, preventing sign errors. Synthetic division only works for polynomials divided by linear expressions with a leading coefficient equal to [latex]1. We start by writing down the coefficients from the dividend and the negative second coefficient of the divisor. Note that we explicitly write out all zero terms!
Bring down the first coefficient and multiply it by the divisor. Now we can also see what the remainder is, just by repeating the procedure:. This way we can still use synthetic division. Finding factors of polynomials is important, since it is always best to work with the simplest version of a polynomial.
When multiplying, things are put together. When factoring, things are pulled apart. Factoring is a critical skill in simplifying functions and solving equations. There are four basic types of factoring. In each case, it is beneficial to start by showing a multiplication problem, and then show how to use factoring to reverse the results of that multiplication. When factoring, this property is done in reverse.
We now divide each term with this common factor to fill in the blanks. Doing this for each term, we obtain:. Other examples are. This type of factoring only works in this specific case: the middle number is something doubled, and the last number is that same value squared.
Furthermore, although the middle term can be either positive or negative, the last term cannot be negative. Both the numerator and denominator have a common factor of 2x. For dividing polynomials by binomials or any other type of polynomials , the most common and general method is the long division method. When there are no common factors between the numerator and the denominator, or if you can't find the factors, you can use the long division process to simplify the expression.
Here, 4x 2 - 5x - 21 is the dividend, and x - 3 is the divisor which is a binomial. Observe the division shown below, followed by the steps.
Step 1. Divide the first term of the dividend 4x 2 by the first term of the divisor x , and put that as the first term in the quotient 4x. Step 2. Multiply the divisor by that answer, place the product 4x 2 - 12x below the dividend. Step 4. Repeat the same process with the new polynomial obtained after subtraction.
Synthetic division is a technique to divide a polynomial with a linear binomial by only considering the values of the coefficients. In this method, we first write the polynomials in the standard form from the highest degree term to the lowest degree terms. While writing in descending powers, use 0's as the coefficients of the missing terms. Follow the steps given below for dividing polynomials using the synthetic division method:.
Step 1: Write the divisor in the form of x - k and write k on the left side of the division. Here, the divisor is x-4, so the value of k is 4. Step 2: Set up the division by writing the coefficients of the dividend on the right and k on the left. Step 3: Now, bring down the coefficient of the highest degree term of the dividend as it is. Here, the leading coefficient is 1 coefficient of x 2. Step 4: Multiply k with that leading coefficient and write the product below the second coefficient from the left side of the dividend.
Step 5: Add the numbers written in the second column. Step 6: Repeat the same process of multiplication of k with the number obtained in step 5 and write the product in the next column to the right.
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