(x^3-6x^2+9)/(x-4) Synthetic Division

3 min read Jun 17, 2024

Synthetic Division: A Step-by-Step Guide for (x^3 - 6x^2 + 9) / (x - 4)

Synthetic division is a simplified method for dividing polynomials, particularly when the divisor is a linear expression (like x - a). Let's walk through the process for the example of (x^3 - 6x^2 + 9) / (x - 4).

Step 1: Set up the division problem

Write the coefficients of the dividend (x^3 - 6x^2 + 9) in a row: 1 -6 0 9 (Notice we include a zero for the missing x term).
Write the constant term of the divisor (x - 4) as a number outside the row: 4

     4 | 1  -6   0   9
        -----------------

Step 2: Bring down the first coefficient

Bring down the first coefficient (1) below the line.

     4 | 1  -6   0   9
        -----------------
         1

Step 3: Multiply and add

Multiply the number you just brought down (1) by the divisor (4) and write the result (4) under the next coefficient (-6).
Add the numbers in the second column (-6 + 4 = -2).

     4 | 1  -6   0   9
        -----------------
         1  -2

Step 4: Repeat the process

Repeat steps 2 and 3 for the remaining coefficients:
- Multiply -2 by 4 and write the result (-8) under the 0.
- Add 0 and -8 to get -8.
- Multiply -8 by 4 and write the result (-32) under the 9.
- Add 9 and -32 to get -23.

     4 | 1  -6   0   9
        -----------------
         1  -2  -8  -23

Step 5: Interpret the result

The last number (-23) is the remainder.
The other numbers (1, -2, -8) are the coefficients of the quotient polynomial. Since the dividend was a cubic polynomial (x^3), the quotient is a quadratic:

Therefore, the result of (x^3 - 6x^2 + 9) / (x - 4) is:

Quotient: x^2 - 2x - 8
Remainder: -23

We can express this as: (x^3 - 6x^2 + 9) / (x - 4) = x^2 - 2x - 8 - 23/(x-4)