Using Synthetic Division to Save Time in Calculus and Algebra

1. Find the Degree of the Polynomial

The degree of a polynomial is equal to the greatest power present in the function. For instance, if 2x³ is the largest power number present in the function, then that polynomial will be considered to have a degree of three.

This value is important because it tells you how many times the polynomial will have to be divided to be completely factored out.

2. Sort by Leading Coefficient

The leading coefficient of a polynomial is the variable of the function attached to the highest power. To perform synthetic division, you first must sort the polynomial by highest coefficients in descending order.

For any coefficient that doesn’t have a corresponding power, insert a zero value. This will be helpful for later steps.

As an example, the polynomial x² + 6 + 3x⁴ + 8x becomes 3x⁴ + 0x³ + x² + 8x + 6 when sorted by leading coefficient.

Now, we our going to take our sorted polynomial and divide it by the linear factor x + 1 using synthetic division. Screenshots in this guide were taken using E Math Help’s synthetic division calculator.

3. Performing Synthetic Division

Take the constant (1) from the divisor (x + 1) and multiply it by -1. Then add it to the left side in a small box or a separated area. This is the number you’ll use for the synthetic division.
Bring down the leading coefficient next to the box.
Multiply the number in the box by the number you just brought down and write the result underneath the second coefficient.
Add the numbers in the second column and write the sum underneath.
Move over to the third column. Repeat the multiplication and addition steps until you’ve gone through all the coefficients. This is the basis of the synthetic division method.
The last number you get is the remainder, and the other numbers form the coefficients of the quotient polynomial. If the polynomial divides perfectly into the linear factor, you’ll end up with zero remainder.

The result will give you the quotient and the remainder of the division. The degree of the quotient polynomial will be one less than the degree of the original polynomial — and just like that, we’ve performed polynomial division.

After we use synthetic division to factor the polynomial 3x⁴ + x² + 8x + 6 divided by (x + 1), our final answer is a lower degree polynomial of 3x³ – 3x² + 4x + 4 and a remainder of 2 divided by (x + 1).

The result probably looks similar to what you would expect using the long division method, but hopefully you find this new method to keep the data much more organized.

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