How to do synthetic division

As students delve deeper into the world of polynomial division, they often encounter the concept of synthetic division. Picture a high schooler, juggling multiple math concepts as he prepares for an upcoming exam, who suddenly realizes he struggles with dividing polynomials quickly. Confused but determined, he decides to search for a clear, straightforward method to master synthetic division, hoping to improve his understanding and boost his grades.

Synthetic division is a simplified method of dividing a polynomial by a binomial of the form (x – c). To perform synthetic division, list the coefficients of the polynomial, then drop the leading coefficient down, multiply it by c, and add it to the next coefficient. Repeat this process across all coefficients to find the quotient and remainder.

To perform synthetic division, follow these steps:

1. Set Up the Synthetic Division: Write down the coefficients of the polynomial you’re dividing. For example, if you’re dividing \(2x^3 + 3x^2 – 5x + 6\) by \(x – 2\), you would use the coefficients [2, 3, -5, 6].

2. Identify the Value: Note the root of the divisor (here, \(x – 2\) gives us \(c = 2\)).

3. Draw the Synthetic Division Bar: Write the coefficients in a row and place the value \(c\) on the left side, separated by a vertical bar.

4. Perform the Division:

– Drop down the leading coefficient (2).

– Multiply this number (2) by \(c\) (2) to get 4, then add it to the next coefficient (3) to yield 7.

– Repeat: multiply 7 by \(c\) (2) to get 14, add to \(-5\) to get 9.

– Lastly, multiply 9 by \(c\) (2) to get 18, add to 6 to get 24.

5. Interpret the Result: The final line of numbers represents the coefficients of the quotient polynomial, with the last number being the remainder. In this case, the result means that \(2x^2 + 7x + 9\) is the quotient and \(24\) is the remainder.

By following these steps, synthetic division can simplify the process of polynomial division, making it easier to understand and quicker to execute.