# Why does polynomial long division work?

• BenB
In summary, polynomial long division is similar to numerical long division, and one method of solving it is through synthetic division. Another approach is to factor the numerator, but if that is not possible, you can still solve it by writing the division as a sum of fractions and dividing accordingly.
BenB
So I'm in a college algebra class and I know how to do polynomial long division. I'm curious as to why polynomial long division works. I've looked at some proofs, but they use scary symbols that I don't understand (I am quite dumb). Do I need very high-level math to comprehend why polynomial long division works? What I'd like to see, if it's possible, is an example of a polynomial division problem being solved with just basic algebra. How would I solve, for example, (x2-x-6)/(x-1) without long division? (sorry, don't know how to use Latex)

polynomial division is very similar to numerical long division.

A common form of polynomial long division is synthetic division:

http://en.wikipedia.org/wiki/Synthetic_division

which may show you how similar they are and why they work.

How would I solve, for example, (x2-x-6)/(x-1) without long division? (sorry, don't know how to use Latex)

Have you tried factoring the numerator?

Last edited by a moderator:
Since neither factor is x- 1, I don't believe factoring helps with the division.

$$\frac{x^2- x}{x- 1}+ \frac{-6}{x- 1}= \frac{x(x- 1)}{x- 1}+ \frac{-6}{x- 1}$$
$$= x+ \frac{-6}{x- 1}$$
so x- 1 divides into $x^2- 1$ x times with remainder -6.

You could also use "synthetic division" as shown here: http://www.purplemath.com/modules/synthdiv.htm

Polynomial long division works because it is based on the fundamental properties of polynomials. In order to understand why it works, we need to first understand what a polynomial is and how it is structured.

A polynomial is an algebraic expression that contains variables and constants, and is made up of terms that are added or subtracted. Each term has a coefficient, which is a number that multiplies the variable(s), and a degree, which is the highest exponent of the variable(s) in that term. For example, the polynomial x^2 - x - 6 has three terms: x^2, -x, and -6. The coefficient of x^2 is 1, the coefficient of -x is -1, and the coefficient of -6 is -6. The degree of x^2 is 2, the degree of -x is 1, and the degree of -6 is 0.

When we divide one polynomial by another, we are essentially trying to find the quotient and remainder of the division. This is similar to dividing numbers, where we find the quotient and remainder when one number is divided by another. However, with polynomials, we have to take into account the structure of the polynomials, specifically their degrees.

In the example you provided, (x^2 - x - 6)/(x - 1), we have a polynomial of degree 2 being divided by a polynomial of degree 1. In order to find the quotient, we need to "cancel out" the highest degree term in the dividend (x^2) by using the appropriate term in the divisor (x). This process is known as "long division" because we are essentially dividing each term of the dividend by the divisor, just like we do in long division with numbers.

To solve this problem without long division, we can use basic algebra. We can factor the numerator and denominator to simplify the expression:

(x^2 - x - 6)/(x - 1) = (x-3)(x+2)/(x-1)

Now, we can see that the x-1 term in the denominator can cancel out with the x-3 term in the numerator, leaving us with the simplified expression of x+2 as the quotient.

In conclusion, polynomial long division works because it is based on the fundamental properties of polynomials and their structure. While it may seem complicated at first, understanding the basic principles of polynomials and their degrees can

## 1. Why is polynomial long division necessary?

Polynomial long division is necessary because it allows us to divide a polynomial by another polynomial, which is a fundamental operation in algebra. This process helps us find the quotient and remainder of a polynomial division, which can be useful in solving equations, graphing, and other applications.

## 2. What is the step-by-step process of polynomial long division?

The step-by-step process of polynomial long division involves dividing the highest degree term of the dividend (numerator) by the highest degree term of the divisor (denominator). The resulting term becomes the first term of the quotient. Then, we multiply the divisor by the first term of the quotient and subtract it from the dividend. This process is repeated until the remaining polynomial has a degree less than the divisor, and the final result is the quotient and remainder of the division.

## 3. How does polynomial long division work?

Polynomial long division works by using the same principles as regular long division. The dividend is divided by the divisor term by term, starting with the highest degree terms. The resulting terms are then multiplied by the divisor and subtracted from the dividend, leaving a new polynomial to be divided. This process is repeated until the remaining polynomial has a lower degree than the divisor, and the final result is the quotient and remainder of the division.

## 4. Why do we need to use coefficients in polynomial long division?

Coefficients are necessary in polynomial long division because they represent the numerical values of the terms in a polynomial. Without coefficients, we would not be able to accurately divide and subtract the terms in the polynomial division process. Coefficients also help us identify the degree of a polynomial and determine the appropriate steps to take in the division process.

## 5. How is polynomial long division related to synthetic division?

Polynomial long division and synthetic division are both methods used to divide polynomials. The main difference is that polynomial long division is a more general method that can handle any type of polynomial division, while synthetic division is a faster and more efficient method that can only be used for dividing by linear binomials (polynomials with two terms). Additionally, synthetic division is based on the same principles as polynomial long division, but it uses a different notation and eliminates the need for writing coefficients.

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