Long division of polynomials is a nightmare. Synthetic division is its fast, elegant, and powerful cousin.

This is a shortcut that works ONLY when you are dividing a polynomial by a simple binomial like '(x - k)'.

Step 1: The Setup. Write down only the coefficients of your polynomial. From '(x - k)', use just the value 'k' and put it in a box.

Step 2: The First Move. Bring down the first coefficient, straight down below the line.

Step 3: The Loop. Multiply the number you just brought down by the 'k' in the box. Write the result up in the next column.

Step 4: The Loop. Add the two numbers in that column straight down.

Step 5: The Loop. Repeat this 'multiply, then add' process until you reach the end of the coefficients.

The final number you get is the Remainder. It should be 0 if '(x - k)' is a true factor.

The other numbers below the line are the coefficients of your new, smaller polynomial answer.

It seems strange at first, but with practice, it's an incredibly fast and efficient tool.