Synthetic division is an algorithm for quickly dividing polynomials when the divisor is in the form x-r. Synthetic division is formally called Ruffini's rule. Synthetic division is commonly used to verify roots of a polynomial.
Step | Equation(s) | Description |
---|---|---|
1 | This is the dividend. | |
2 | This is the divisor. | |
3 | This is the operation that will be performed. | |
4 | Identify the implied coefficients and fill them in. Applying the property of multiplying by 1 to the term x^{3} gives an implied coefficient is 1. Notice that there is no term x^{2}. Applying the property of multiplying by 0 then the property of adding 0 to the x^{2} term give 0·x^{2}. | |
5 | Copy the coefficients into the top of a grid. | |
6 | Copy the additive inverse of the constant term of the divisor into the grid on the left. | |
7 | Copy down the coefficient of the term with the largest degree. In this case, the coefficient is 1. | |
8 | Multiply the first number on the bottom by the number on the left. Place the result in the 2^{nd} row, the next column to the right. | |
9 | Add the numbers in the first two rows of the next column. Place the result in the 3^{rd} row. | |
10 | Multiply the number on the 3^{rd} row of the current column by the number on the left. Place the result in the 2^{nd} row, the next column to the right. | |
11 | Add the numbers in the first two rows of the next column. Place the result in the 3^{rd} row. | |
12 | Multiply the number on the 3^{rd} row of the current column by the number on the left. Place the result in the 2^{nd} row, the next column to the right. | |
13 | Add the numbers in the first two rows of the next column. Place the result in the 3^{rd} row. | |
14 | Multiply the number on the 3^{rd} row of the current column by the number on the left. Place the result in the 2^{nd} row, the next column to the right. | |
15 | Copy the coefficients of the result into a polynomial. | |
16 | Simplify the polynomial. | |
17 | Here is the problem with the result. | |
18 | Check the work by multiplying the result by the divisor. | |
Example 1 |
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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