Synthetic Division

Definition

Synthetic division is a simplified method of dividing a polynomial by a binomial of the form x - c, where c is a constant. It is a quicker and more straightforward alternative to the traditional long division of polynomials, especially when the divisor is a linear polynomial. Synthetic division is particularly useful for finding the roots of polynomials and for simplifying polynomial division when the divisor is of the form x - c.

Procedure for Synthetic Division

1. Set Up the Division:

   Write down the coefficients of the dividend polynomial in descending order of degree. If any term is missing, use a coefficient of 0 for the missing degree. Write the constant c (from the divisor x - c) to the left side of the synthetic division setup.

2. Bring Down the First Coefficient:

   Bring down the first coefficient (the leading coefficient) of the dividend polynomial directly to the bottom row.

3. Multiply and Add:

   Multiply the constant c by the number just written in the bottom row, and write the result under the next coefficient in the top row. Add the result to the coefficient above it, and write the sum in the bottom row.

4. Repeat the Process:

   Continue multiplying the constant c by the last number written in the bottom row and adding the result to the next coefficient above until you have processed all coefficients.

5. Interpret the Result:

   The numbers in the bottom row represent the coefficients of the quotient polynomial, with the last number being the remainder.

Example 1:
Divide 2x3 - 6x2 + 2x - 4 by x - 2

Step 1: Set Up the Division

Dividend coefficients: 2, -6, 2, -4
Divisor: x - 2, so c = 2

2 |  2  -6  2  -4
  |      4 -4 -4
   -----------------
  |  2  -2 -2  -8
            

Step 2: Bring Down the First Coefficient

Bring down the 2 (the leading coefficient).

Step 3: Multiply and Add

• Multiply 2 × 2 = 4 and write it under the next coefficient -6. Add: -6 + 4 = -2.

• Multiply 2 × - 2 = - 4 and write it under the next coefficient 2. Add: 2 - 4 = -2.

• Multiply 2 × - 2 = - 4 and write it under the next coefficient -4. Add: -4 - 4 = -8.

Step 4: Interpret the Result

The quotient is 2x2 - 2x - 2, and the remainder is -8.

Thus, the division yields:
$$ \frac{2x^3 -6x^2 +2x -4}{x -2} $$ $$ = 2x^2 - 2x -2 $$ with a remainder of -8

Example 2: Divide x3 + 4x2 - 3x + 2 by x + 1

Step 1: Set Up the Division

Dividend coefficients: 1, 4, -3, 2
Divisor: x + 1, so c = -1

-1 |  1   4  -3   2
   |     -1  -3   6
    -----------------
   |  1   3  -6   8
            

Step 2: Bring Down the First Coefficient

Bring down the 1 (the leading coefficient).

Step 3: Multiply and Add

• Multiply -1 × 1 = -1 and write it under the next coefficient 4. Add: 4 + (-1) = 3.

• Multiply -1 × 3 = -3 and write it under the next coefficient -3. Add: -3 + (-3) = -6.

• Multiply -1 × -6 = 6 and write it under the next coefficient 2. Add: 2 + (6) = 8.

Step 4: Interpret the Result

The quotient is x2 + 3x - 6, and the remainder is 8.

Thus, the division yields:

$$ \frac{ x^3 + 4x^2 -3x +2}{x + 1} $$ $$ = x^2 + 3x -6 $$

with a remainder of 8

Or written as

$$ x^3 + 4x^2 -3x +2 = (x + 1)(x^2 + 3x -6) + 8 $$