Synthetic Division Calculator

What Is Synthetic Division?

Synthetic division is a shorthand, algorithmic method for dividing a polynomial by a linear binomial of the form `(x - c)`. It is a faster and less cumbersome alternative to traditional polynomial long division. The technique simplifies the division process by focusing only on the coefficients of the polynomial, making it an invaluable tool in algebra for finding roots (or zeros) of polynomials and for evaluating polynomials at a specific value. This synthetic division calculator not only provides the answer but also illustrates the step-by-step process, making it a powerful learning aid.

How to Perform Synthetic Division

The process follows a simple, repeatable algorithm. The result of the division is expressed in the form:

Let's divide the polynomial `P(x) = x³ - 2x² - 5x + 6` by `(x - 1)` (so, c = 1).

1. Set up the Division: Write the value of 'c' to the left. To its right, list all the coefficients of the dividend polynomial. Make sure to include a '0' for any missing terms (e.g., for `x³ + x - 5`, the coefficients would be 1, 0, 1, -5).
2. Bring Down the First Coefficient: Drop the leading coefficient straight down below the line.
3. Multiply and Add: Multiply the number you just brought down by 'c'. Write the result under the next coefficient. Add the two numbers in that column together and write the sum below the line.
4. Repeat: Continue the "multiply and add" process for all remaining coefficients.
5. Interpret the Result: The last number below the line is the remainder. The other numbers are the coefficients of the quotient polynomial, whose degree is one less than the original dividend.

In our example, the final numbers would be `1, -1, -6, 0`. This means the quotient is `1x² - 1x - 6` and the remainder is 0.

Applications of Synthetic Division

Synthetic division is a cornerstone of the Factor Theorem and the Remainder Theorem, making it incredibly useful in algebra.

• Finding Roots of Polynomials: According to the Factor Theorem, if the remainder of `P(x) / (x - c)` is zero, then `(x - c)` is a factor of `P(x)`, and `c` is a root of the equation `P(x) = 0`. This is the most common application of synthetic division. After finding one root, you can continue to find others by factoring the resulting quotient. For quadratic quotients, our Quadratic Formula Calculator can find the remaining roots.
• Evaluating Polynomials: The Remainder Theorem states that the remainder obtained from dividing `P(x)` by `(x - c)` is equal to `P(c)`. This means you can use synthetic division as a quick way to evaluate a polynomial at a specific point `x=c`.

For a deeper understanding of polynomial functions and their properties, academic resources like Math is Fun provide excellent, easy-to-follow tutorials.