Synthetic Division

 Math can be very complex in some cases. The complexity of the mathematic language means specific terms are needed to distinguish between certain kinds of values, functions, and equations.

Polynomials are one example of a very specific type of equation. This kind of equation contains only variables and coefficients of positive integers. Also, only the functions of addition, subtraction, and multiplication. These equations, or statements, can look intimidated. However, they can be reduced and simplified using synthetic division.

An example of a polynomial with a single variable could look like this, x2 − 4x + 7. An example of a polynomial with multiple variables could look like this, x3 + 2xyz2 − yz + 1. The two look very different, but they are fundamentally the same.

Synthetic division is a shortcut for finding the zeros within the statement and effectively reducing it to a more easily solved statement.

Solving With Synthetic Division

There are two types of synthetic division. Regular synthetic division uses fewer steps, but may not be easier. The difference is in how the functions are carried out. To show the process, an example problem is needed. 2x + 4 / 2 is an example of a single variable polynomial.

This problem is solved by simplifying and reducing the functions. 2x + 4 / 2 = 2x /2 + 4 / 2, following the rule of functions will produce the result of x + 2.

Typed math problems look very different from the statements themselves. When using mathematic language the problem used above would look like this,

The thing about polynomials is that there may be more than one solution, but the answer is always the same.

The point of synthetic division is to reduce the statement to its most simple problem. Solving for known problems in the statement by following the rule of functions will yield a more solvable problem.

It’s important to note at this point that x in the problem above cannot equal zero. This is because the original problem called for division of x and division by zero is not allowed in math. It is a nonfunction and will yield no result since there is no function to be completed.

The second example shows extended synthetic division. The difference is subtle, but, extended synthetic division is used primarily for statements with multiple variables such as  y = x2 + 5x + 6. By solving the statement diagonally, the problem becomes solvable and the values of both variables can be found.

To make the problem easier to read, compact extended can be used. This method presents each step and allows information to flow.

Any of the three methods may be used depending on the number of variables. The functions and answers will remain the same, though. The principals of this type of solution are found in the Euclidian algorithm. Which will provide a longer and equally reliable way to solve polynomials with any number of variables.