(-2x^3+5x^2-x+2)/(x+2) Synthetic Division

Jasonbradley

3 min read

·

Jun 16, 2024

41

Share

Solving Polynomial Division Using Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x - a). It's a more efficient way to perform long division, especially when the divisor is a linear binomial. Let's illustrate this with an example.

Example: Dividing (-2x^3 + 5x^2 - x + 2) by (x + 2)

1. Set up the problem:

• Write the coefficients of the dividend, (-2x^3 + 5x^2 - x + 2), in a row: -2 5 -1 2

• Since we're dividing by (x + 2), write -2 to the left of the coefficients.

    -2 | -2  5  -1  2
  

2. Bring down the first coefficient:

• Bring down the first coefficient, -2, below the line.

    -2 | -2  5  -1  2
       -2
  

3. Multiply and add:

• Multiply the number you just brought down (-2) by the divisor (-2): (-2) * (-2) = 4. Write this result under the next coefficient (5).

• Add the numbers in the second column: 5 + 4 = 9.

    -2 | -2  5  -1  2
       -2
    -------
       -2  9
  

4. Repeat steps 3 and 4:

• Multiply the new number (9) by the divisor (-2): 9 * (-2) = -18. Write this result under the next coefficient (-1).

• Add the numbers in the third column: -1 + (-18) = -19.

    -2 | -2  5  -1  2
       -2  -18
    -------
       -2  9  -19
  

• Repeat this process for the last coefficient. Multiply -19 by -2 to get 38. Add this to 2 to get 40.

    -2 | -2  5  -1  2
       -2  -18  38
    -------
       -2  9  -19  40
  

5. Interpret the result:

• The numbers below the line represent the coefficients of the quotient, starting with the highest power of x: -2x^2 + 9x - 19.
• The last number, 40, is the remainder.

Therefore, the result of dividing (-2x^3 + 5x^2 - x + 2) by (x + 2) is:

-2x^2 + 9x - 19 + 40/(x + 2)