Synthetic Division to Divide Polynomials
This tutorial will teach you how to use synthetic division – an extremely useful tool when dealing with polynomials. It is a simplified version of long division that allows you to divide polynomials with ease.
The first step in synthetic division is to determine if synthetic division can be used given the two polynomials you need to divide. The dividend (the polynomial which is being divided, x^2+7x–11 in this case) can be any polynomial with a degree of one or greater, meaning the highest exponent in the polynomial must be at least 1. The divisor (the polynomial divided from the dividend, x–1 in this case) must have a degree of 1. This means it must be in the form of x-c where c is any rational number.
If synthetic division can be performed, the next step is to set up the polynomials. This is done by first writing the c value from the divisor (which is in the form x-c). Then, a corner is drawn around this number, it should look like a box around the number with only the bottom and right-hand side of the box. The dividend is written to the right of this number and corner on the same line. Only the coefficients of the dividend should be written. Finally, draw a line under all of the numbers, skipping a line. In this case, it should look like this
1 ] 1 7 -11 –––––––––––––––
Bring down the first coefficient from the dividend directly below the line you drew
1 ] 1 7 -11 ––––––––––––––– 1
Multiply the rightmost number on the bottom row by the c value from the divisor and write the product under the next coefficient. Then, add the terms in that column. Repeat this process until all columns are filled. That should look like this:
1 ] 1 7 -11 1 ––––––––––––––– 1 1 ] 1 7 -11 1 ––––––––––––––– 1 8 1 ] 1 7 -11 1 8 ––––––––––––––– 1 8 1 ] 1 7 -11 1 8 ––––––––––––––– 1 8 3
The last step is to write your answer using the bottom row you created. The rightmost number on the bottom row is your remainder (it may be zero). The numbers before that are your resulting polynomial in order of descending degree. In this case, you have the numbers 1 and 8, meaning the polynomial is 1x+8 and the remainder is 3. The remainder is written as a fraction added to the resulting polynomial with the numerator being the remainder and the denominator being the divisor. This makes your final answer:
x + 8 + 3/(x-1)
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