How to Calculate Synthetic Division
Synthetic division is a shorthand, or shortcut method, of polynomial division in the special case of dividing by a linear factor — and it only works in this case. Synthetic division is generally used for finding roots (or zeroes) of polynomials instead of using the traditional long division method.
- P(x): The polynomial you are dividing. Input its coefficients in order of descending degree.
- Divisor (c): To divide by (x - 2), use c = 2. To divide by (x + 3), use c = -3.
- Quotient: The resulting polynomial, which will always be one degree lower than P(x) (a "depressed polynomial").
- Remainder: If the remainder is 0, then (x - c) is a perfect factor of P(x).
| Feature | Polynomial Long Division | Synthetic Division |
|---|---|---|
| Divisor Types | Works for dividing by polynomials of any degree. | Only works when dividing by linear binomials (x - c). |
| Setup Speed | Slower, requires writing out full variables and exponents. | Faster, uses only the numeric coefficients. |
| Calculation Process | Involves continuous subtraction down the line. | Involves continuous addition and multiplication. |
| Visual Space | Takes up significant page vertical space. | Compact, typically fitting within 3 horizontal rows. |
Frequently Asked Questions
What is the Remainder Theorem?
The Remainder Theorem states that if a polynomial P(x) is divided by (x - c), the remainder of that division is exactly equal to P(c). This means synthetic division is also a rapid way to evaluate a polynomial at a specific value.
What happens if a term is missing in my polynomial?
When setting up synthetic division, you must account for every descending power of x. If a term is missing, such as the x² term in x³ - 4x + 5, you must include a zero as a placeholder in your coefficients (e.g., 1, 0, -4, 5). Failing to include the zero will result in an incorrect calculation.
Can synthetic division be used to factor a polynomial?
Yes. If you perform synthetic division using a specific root and the remainder is zero, you have proven that (x - c) is a factor. You can then continue to factor the resulting "depressed polynomial" (the quotient) to find the remaining roots.