Why does polynomial long division work?

[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)

 

[jedishrfu]

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.

 

[SteamKing]

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?

 

[HallsofIvy]

Since neither factor is x- 1, I don't believe factoring helps with the division.

Instead write this as
$$\frac{x^2- x}{x- 1}+ \frac{-6}{x- 1}= \frac{x(x- 1)}{x- 1}+ \frac{-6}{x- 1}$$
$$= x+ \frac{-6}{x- 1}$$
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

 

[CellCoree]

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