how to do synthetic division

Synthetic division is a shortcut method used to divide a polynomial by a linear factor of the form x−ax-ax−a. It is simpler and faster than traditional long division, especially when the divisor is linear and its leading coefficient is 1. Here is how to do synthetic division step-by-step:

Steps to Perform Synthetic Division

1. Write the coefficients of the dividend polynomial in descending order of powers. Include zeros for any missing powers.
2. Identify the zero of the divisor : For a divisor x−ax-ax−a, the zero is aaa. This value is used in the synthetic division process.
3. Set up the synthetic division tableau :
   • Write the zero of the divisor to the left.
   • Write the coefficients of the dividend in a row to the right.
4. Bring down the first coefficient (the leading coefficient) directly to the bottom row.
5. Multiply the zero of the divisor by the number just written in the bottom row, and write the result under the next coefficient.
6. Add the column of numbers (the coefficient and the product) and write the sum in the bottom row.
7. Repeat the multiply-and-add process for each coefficient until all coefficients have been processed.
8. Interpret the result :
   • The numbers in the bottom row (except the last one) are the coefficients of the quotient polynomial, which will be one degree less than the original polynomial.
   • The last number in the bottom row is the remainder.

Example

Divide 2x3−3x2+4x+52x^3-3x^2+4x+52x3−3x2+4x+5 by x+2x+2x+2:

• The zero of the divisor x+2x+2x+2 is −2-2−2.
• Coefficients of the dividend: 2, -3, 4, 5.

Set up:

-2 |  2   -3   4   5
     |      -4  14 -36
     -----------------
       2   -7  18  -31

• Bring down 2.
• Multiply 2 by -2 = -4, add to -3 = -7.
• Multiply -7 by -2 = 14, add to 4 = 18.
• Multiply 18 by -2 = -36, add to 5 = -31.

Quotient: 2x2−7x+182x^2-7x+182x2−7x+18, remainder: −31-31−31. So,

2x3−3x2+4x+5x+2=2x2−7x+18+−31x+2\frac{2x^3-3x^2+4x+5}{x+2}=2x^2-7x+18+\frac{-31}{x+2}x+22x3−3x2+4x+5​=2x2−7x+18+x+2−31​

Important Notes

• Synthetic division works only when dividing by a linear factor x−ax-ax−a.
• The leading coefficient of the divisor should be 1. If not, divide the divisor by its leading coefficient first.
• It replaces division and subtraction in long division with multiplication and addition, making calculations simpler.
• The quotient polynomial degree is one less than the dividend polynomial.

This method is efficient and less error-prone than long division for linear divisors