Long Division and Synthetic Division | The Remainder Theorem | The Rational Roots Theorem

Long Division and Synthetic Division

Another way to divide a polynomial by another polynomial is to use synthetic division. Really synthetic division does the same things that are done in the long division.
 
To show this similarity, we will show a problem being done by long division next to the same problem being done with synthetic division.

Long Division Steps	Long Division Explanation	Synthetic Division Steps	Synthetic Division Explanation
	Given Problem. It is very important that when you do a long division problem of this type that you write the dividend in descending order. And if a degree is missing in the sequence, you need to leave space for this missing term. Usually this is done by writing the variable with the appropriate exponent and a zero coefficient.		In synthetic division, the coefficient of the variable must be one, so the first step in this problem is to divide the divisor and the dividend by 3. This makes the dividend coefficients what you see in the top row. For the divisor, we have (x+2/3). What value of x would make the divisor equal to zero? A value of x=-2/3. This is the problem constant.
	In your division, you must think division of monomials. So here you should be thinking, "What times 3x is 6x cubed?" - Answer: 2x squared. So now multiply (3x+2) times 2x squared, subtract, and bring down the next term.		Our first step is to bring down the first coefficient (2).
	In your division, you must think division of monomials. "What times 3x is -9x squared?" - Answer: -3x. So now multiply (3x+2) times -3x, subtract and bring down next term.		We then take this 2 times our constant of (-2/3) and add this to our 2nd column. The result is -3.
	"What times 3x is -6x?" - Answer: -2. Now multiply (3x+2) times -2 and subtract. There is no remainder.		We then take this -3 times our constant of (-2/3) and add this our 3rd column. The result is -2.
	Multiply the divisor by the quotient and see if it checks with the dividend. It does.		We then take this -2 times our constant of (-2/3) and add this to our 4th column. The result is zero. The bottom row of this result tells us the coefficients of the answer as seen at the left. The last number (0) tells us the remainder.

The Remainder Theorem

A great time saver is the Remainder Theorem. This theorem allows you to find out (without doing long or synthetic division) if a number is a solution to a polynomial equation or not. The theorem says that to find out if (x-s) is a solution to the polynomial function f (x), just check the value of f (s). The value of f (s) is the remainder after doing long division. If this remainder is zero, then s is a solution, and (x-s) is a factor of the polynomial.

Example:

Check if -4 is a solution to	Sample Problem
	Find f (-4). This value will be the remainder after dividing the polynomial by (x+4).
	We see that  f (-4)=0. Therefore (x+4) divides evenly into the polynomial and is therefore a factor!
	Here we check this by using long division. We see that (x+4) really is a factor of the function, and that when divided into the function, really does result in zero remainder.
	We could even use this new ability to save time on long division to help us factor polynomials of degree greater than two. The above problem could have been continued to factor the original 3rd degree polynomial.

The Remainder Theorem @ PurpleMath

The Rational Roots Theorem.

Even with the remainder theorem, we still have to pick which numbers might be the root, and then run them through the remainder theorem to see if there is a remainder or not. Another tool that can help us narrow down the choices we make in "trying them out" using the remainder theorem is the Rational Roots Theorem. The Rational Roots Theorem narrows down the attempts for finding the rational roots. We still have a problem because there may be no rational roots, but it allows us to try a smaller number of roots.

Here is how the Rational Roots Theorem works. Let's work through an example:

	Find all of the possible rational roots for this function.
	First, we make sure that the function is written in descending order (from largest degree term to the least). P is identified as the coefficient of the constant term, Q is identified as the coefficient of the highest degree term.
	Next, find all of the integer factors of P, and all of the integer factors of Q. Now we find all of the possible P/Q combinations. These are the candidates for being rational roots. The next step (although not asked for in this problem) is to check each rational root candidate using the Remainder Theorem.

The Rational Roots Theorem @ Purple Math

Synthetic Division @ Purple Math