Synthetic Division: A Simplified Approach to Polynomial Division
Synthetic division is an efficient method for dividing polynomials, particularly when the divisor is a linear factor. This page provides a comprehensive overview of the technique, complete with synthetic division of polynomials example problems with solutions.
Definition: Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x-a) or (x+a).
The process of synthetic division involves the following steps:
- Arrange the coefficients of the dividend polynomial in descending order of degree.
- Write the constant term of the divisor (with the opposite sign) in a box.
- Bring down the first coefficient.
- Multiply the result by the number in the box and add it to the next coefficient.
- Repeat step 4 until all coefficients have been used.
Example: For (x²+x-2) ÷ (x-1), we use 1 in the box (since the divisor is x-1).
The solution is presented as follows:
1 | 1 1 -2
| 1 2
+-----------
1 2 0
The result shows that (x²+x-2) = (x-1)(x+2) + 0.
Highlight: If the remainder is 0, the polynomial has been factored.
The page also includes more complex examples, such as dividing a higher-degree polynomial:
Example: (5x³-6x²+3x+11) ÷ (x-2)
This example demonstrates how synthetic division can handle polynomials of higher degrees efficiently.
Vocabulary: Dividend - the polynomial being divided; Divisor - the polynomial by which we are dividing; Quotient - the result of the division; Remainder - what's left over after division.
The page concludes with examples of using synthetic division for polynomials with missing terms, emphasizing the importance of including zero coefficients for absent terms.
Highlight: When using synthetic division of polynomials, always include zero coefficients for missing terms to maintain the correct degree structure of the polynomial.
This comprehensive guide provides a solid foundation for understanding and applying synthetic division to various polynomial problems, making it an invaluable resource for students learning advanced algebraic techniques.
