Synthetic division is a method for dividing polynomials that is faster and more efficient than long division. This technique is particularly useful when dividing a polynomial by a linear factor. The process involves setting up a compact arrangement of the coefficients and performing simple arithmetic operations.
Highlight: Synthetic division is a streamlined method for polynomial division, especially effective when dividing by linear factors (x - a).
The method follows these general steps:
- Arrange the polynomial in descending order of degree
- Set up the division using only the coefficients
- Bring down the first coefficient
- Multiply the result by the divisor and add to the next coefficient
- Repeat step 4 until all coefficients are processed
Example: For (x³-2x²-5x+6) ÷ (x-3), the synthetic division process yields x²+3x+1 as the quotient with a remainder of 0.
Synthetic division not only simplifies the division process but also helps in finding roots of polynomials and factoring higher-degree polynomials.
Vocabulary: Coefficients are the numerical factors of the variables in a polynomial expression.
This method is an essential tool in algebra and calculus, providing a quick way to perform polynomial division and analyze polynomial behavior.
