fresh_42 said:It is difficult to display here and of course you can always multiply to check the result. I once explained it here:
https://www.physicsforums.com/threa...r-a-polynomial-over-z-z3.889140/#post-5595083Maybe this helps. It is the same as a normal division, step by step.
fresh_42 said:##1 : (1+y^2) = 1 \longrightarrow \Delta = -y^2##
##-y^2 : (1+y^2) = -y^2 \longrightarrow \Delta = y^4##
##y^4 : (1+y^2) = y^4 \longrightarrow \Delta = -y^6##
etc.
There is no restriction what has to be divided by what. In this case we start with ##1/1## and always divide by ##1##.
That's what is more elaborated in the first example I gave you. The deltas are the differences, because long division is a sequence of subtractions.SamRoss said:What is meant by ##1 \longrightarrow \Delta ## and all the other ## \longrightarrow \Delta ## steps?
Polynomial long division is a method used to divide two polynomials, which are algebraic expressions made up of variables and coefficients. It is similar to long division with whole numbers, but instead of dividing numbers, we divide polynomials.
Polynomial long division is important because it allows us to simplify complex polynomial expressions and solve equations involving polynomials. It is also a necessary skill for many advanced mathematical concepts and real-world applications.
To perform polynomial long division, we follow a set of steps:1. Arrange the terms of the dividend and divisor in descending order of their exponents.2. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.3. Multiply the first term of the divisor by the first term of the quotient and subtract it from the dividend.4. Bring down the next term of the dividend and repeat the process until all terms have been brought down.5. The final result is the quotient, with any remainder written over the divisor.
Some common mistakes made in polynomial long division include:- Forgetting to bring down the next term of the dividend.- Making a mistake in the multiplication or subtraction step.- Dividing the wrong terms of the dividend and divisor.- Forgetting to include a placeholder for missing terms in the dividend.- Writing the final result in the wrong format.
You can check your answer in polynomial long division by multiplying the quotient and divisor and adding the remainder. The result should be equal to the dividend. If it is not, then there may be a mistake in your division process. You can also use a calculator to verify your answer.