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# Synthetic Division (Extended Version)

- Author:
- A B Cron

This applet will enable you to learn synthetic division with ease and efficiency. Polynomials of up to degree 6 can be entered, the program will do the arithmetic, you need to learn how to do this on your own. Have fun!!
Enter f(x) into the input box, then enter you first divisor under Trial1. If you get a zero for the remainder of the first divisor or the actual remaider, you then can add a second divisor in A6 under Trial2, repeat until you can go no longer. If you need assistance for this, see this lesson. If your polynomial has degree 2, the quadratic formula or factoring will be more efficient.
If any row has values only the last two columns, you can solve the linear equation for its value of x by setting a x + c = 0.
To determine the values for the divisors in the A column, use the pq estimates of these values. 'p' represents all the factors of the constant term. 'q' represents all factors of the leading coefficient. See this lesson for assistance.
To preserve the formulae in the cells, be sure to type only in the blue or pink boxes. A Reset button will allow you to start an new expression. Use the recycle icon on the upper right of the graphic screen if you need to restore formulae. Sample1 show a fully factorable expression; Sample2 shows a partially factorable expression.
Warning: This applet does not solve all Synthetic Division problems, it will not solve problems with complex roots. It is a tool to help you to learn to do synthetic division yourself; with repeated use you may be able to find constant factors.

Possible Factors
The leading coefficient, q, and the constant term, p, can be used to find possible integer factors for use in Synthetic Division.
Boundaries
Upper and Lower Bound Rules
Let f(x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f(x) is divided by x - h using synthetic division.
If h>0 and each number in the last row is either positive or zero, h is an upper bound for the real zeros of f(x).
If h<0 and the numbers in the last row are alternately positive and negative (zero entries count as positive or negative), h is a lower bound for the real zeros of f(x).
The Intermediate Value Theorem can be used to determine the boundaries.
Positive/Negative Roots
Decartes Rule of Signs This will determine the number of possible positive and negative real solutions.
Rene Descartes came up with a rule of signs that can be used to find the number of possible positive and negative solutions to a polynomial equation.
Positive Roots
The rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or less than it by a multiple of 2. Multiple roots of the same value are counted separately.
Negative Roots
As a corollary of the rule, the number of negative roots is the number of sign changes after negating the coefficients of odd-power terms (otherwise seen as substituting the negation of the variable for the variable itself), or fewer than it by a multiple of 2. {Wikipedia}
I.e., examine the number of sign changes in f(x) for possible positive roots {you can count the sign changes for f(x) line B2-H2 above} and
then examine the number of sign changes in f(-x) for possible negative roots.
This applet can be used by teachers in a demonstration mode in the classroom or teachers can have students load it on their own computers as a worksheet to be completed for a grade.