Synthetic Division and Long Division

Synthetic division and long division can be used for dividing, factoring, and finding the zeros of polynomials. 

Long division of polynomials is basically the same as long division of numbers. You start with a polynomial, say x^4-2x^3+x-4. Lets say you were dividing it by x+2. You set the two polynomials up as you would a basic long division problem, with the dividend under the division sign and the divisor to the left. Make sure to put in a placeholder if a certain degree is missing. In this case, you would add 0x^2.

LaTex leaves a blank space, but just imagine a 0x^2 there. The first step is to look at the first term in the dividend, in this case, x^4. You then look at the first term in the divisor, x. Determine how many times it goes into the first dividend term and write it above the term, matching exponents. For instance, you determine that x goes into x^4 x^3 times, so you write x^3 above 2x^3. Then you have to multiply any other terms in the divisor by x^3 as well, so you have 2x^3. Subtract the entire term (x^4+2x^3) from the first two terms, being sure to distribute the negative.

You then repeat the entire process again with the remainder until you can no longer multiply the divisor by a whole number to factor into the remainder. In this example, you end up with x^3-4x^2+8x-15, with a remainder of 26.

Synthetic division does the same thing, just by a different method. You begin by writing out all of the coefficients of the polynomial in order, leaving a zero as a placeholder for any missing exponents. To the left, write the zero of the term you are dividing by. For instance, we're dividing by x+2, so if x+2=0, then the zero would be -2, so we have -2 to the left.

Start by dropping the first term straight down to below the line. Then multiply that term by the number to the left, and write that below the second number. Add the two together and write the sum below the line. Then continue to multiply that number by the number to the left, writing it under the next number, and adding until you run out of numbers. The resulting numbers are the coefficients of the quotient, each with x to 1 less degree than the coefficient above it. For instance, we have x^3-4x^2+8x-15, with remainder 26. This is the exact same problem as before, just solved with a different method.

Synthetic division can also be used to factor polynomials. Say you have the polynomial x^3-x^2-4x+4. You can set up an equation, dividing that by one of its zeros. You try 1.

You know that factoring a polynomial has worked if you end up with a zero at the end, meaning there is no remainder. You can now rewrite the expression as (x-1)(x^2-4). You can even take it one step further and factor your result. This time you divide it by 2.

It worked again! So now you have a completely factored polynomial that can be written as (x+2)(x-1)(x-2). But how do you know which numbers to try dividing by? There are infinitely many possibilities, after all. One way to narrow down the options is to divide all of the factors of the last term by all the factors of the first term. In that last polynomial, that would leave you with 4,-4,1,-1,2, and -2. They aren't necessarily all zeros, but it considerably narrows down the options.