Synthetic Division

Example Problem for synthetic division: f(x) = 4x⁴ + 3x² – 2x + 7; evaluate f(½) and find the quotient and remainder when f(x) is divided by x – ½ . Note, by the Remainder Theorem, we do the same thing to answer both questions.

1.) Write down the coefficients of the polynomial in descending order, making sure to put zeros in where needed.

Example: There is no x³ in our example problem, so the coefficient of x³ is zero.

Thus we write: 4 0 3 -2 7 for our coefficients.

2.) Write down your value of c. You can calculate f(c) by doing synthitic division and looking at the remainder.

Example: ½

My way: Book’s way:

3.) Bring down the first number:

Example: 4

My way: Book’s way:

4.) Multiply that number by c and add it to the next number:

Example: 4(½) = 2, 2 + 0 = 2

My way: Book’s way:

5.) Repeat the process until we get to the end.

Example: 2(½) = 1, 1 + 3 = 4

My way: Book’s way:

Next: 4(½) = 2, 2 + -2 = 0

My way: Book’s way:

Next: 0(½) = 0, 0 + 7 = 7

My way: Book’s way:

6.) Now you know that f(½) = 7 and f(x)/(x – ½) = 4x³ + 2x² + 4x with remainder 7. I like to leave out the line where the book writes the result of the multiplication and do that part in my head or on a calculator. That makes it easier to use the same chart to do synthetic division by another number. For example if I also wanted to evaluate f(¼),

f(-¼), and f(-½), I could add these rows to the same chart.

Example:

So f(¼) = 429/64, f(-¼) = 493/64, and f(-½) = 9.

Don’t forget if you have a TI-82 or 83, I have a program written up that you can plug into your calculator to do synthetic division.