%2F(x-4)-synthetic-division.jpeg "(x^3-13x-12)/(x-4) Synthetic Division")
Performing Synthetic Division: (x³ - 13x - 12) / (x - 4)
Synthetic division is a shortcut method for dividing polynomials, especially when the divisor is a linear expression of the form (x - a). Let's illustrate how to use synthetic division to divide (x³ - 13x - 12) by (x - 4).
Steps:
-
Set up the division:
- Write the coefficients of the dividend (x³ - 13x - 12) in a row, including any missing terms with a coefficient of 0.
- Write the constant term of the divisor (x - 4) to the left of the coefficients.
- Draw a horizontal line below the coefficients.
4 | 1 0 -13 -12 ------------------ -
Bring down the leading coefficient:
- Bring down the first coefficient (1) below the line.
4 | 1 0 -13 -12 ------------------ 1 -
Multiply and add:
- Multiply the number you just brought down (1) by the divisor's constant term (4).
- Write the product (4) below the next coefficient (0).
- Add the two numbers (0 + 4 = 4) and write the sum below the line.
4 | 1 0 -13 -12 ------------------ 1 4 -
Repeat the process:
- Repeat steps 3 and 4 for each remaining coefficient.
4 | 1 0 -13 -12 ------------------ 1 4 3 4 | 1 0 -13 -12 ------------------ 1 4 3 -4 -
Interpret the results:
- The numbers below the line represent the coefficients of the quotient polynomial, starting with the term of highest degree.
- The last number is the remainder.
Therefore, the result of the division (x³ - 13x - 12) / (x - 4) is:
Quotient: x² + 4x + 3 Remainder: -4
This can also be expressed as:
(x³ - 13x - 12) / (x - 4) = x² + 4x + 3 - 4/(x - 4)
Conclusion
Synthetic division provides a concise and efficient way to divide polynomials, particularly when the divisor is linear. By following the simple steps outlined above, you can quickly obtain the quotient and remainder of the division.