Dividing polynomials can be a lengthy process using the traditional long division method. However, for the special case of dividing a polynomial by a linear binomial of the form (x - c), there is a much faster and more efficient algorithm called synthetic division. It's a powerful shortcut that allows you to find the quotient and remainder with far fewer steps. Our Synthetic Division Calculator is a tool designed to help students learn and master this technique, providing a quick way to check answers and understand the process.
How to Use the Synthetic Division Calculator
Using our calculator to perform synthetic division is easy:
- Enter Polynomial Coefficients: Input the coefficients of your dividend polynomial, starting with the term of the highest degree. Separate them with commas or spaces. Crucially, you must enter a '0' for any missing terms.
- Enter the Root 'c': For a divisor of the form (x - c), input the value of 'c'. Note that if your divisor is (x + 3), your 'c' value would be -3.
- Calculate: Click the "Divide" button.
- View the Result: The calculator will instantly display the coefficients of the resulting quotient polynomial and the final remainder.
The Synthetic Division Process: A Step-by-Step Guide
Synthetic division is a streamlined version of long division that gets rid of the variables and focuses only on the numerical coefficients. Let's walk through an example: divide (x³ - 12x² + 42) by (x - 3).
- Set Up the Problem:
- Write down the value of 'c' from your divisor (x - c). In this case, c = 3.
- To the right, write down the coefficients of the dividend in a row. Remember to include a zero for any missing terms. Our dividend is x³ - 12x² + 0x + 42, so our coefficients are 1, -12, 0, 42.
- Draw a line below the coefficients.
- Bring Down the First Coefficient: Drop the first coefficient (1) straight down below the line.
- Multiply and Add: This is the repeating cycle.
- Multiply: Multiply the number you just brought down (1) by your 'c' value (3). Write the result (3) under the next coefficient (-12).
- Add: Add the numbers in that column (-12 + 3 = -9). Write the sum below the line.
- Repeat the Cycle: Repeat the "Multiply and Add" step until you reach the end.
- Multiply -9 by 3 to get -27. Write it under the 0. Add 0 + (-27) = -27.
- Multiply -27 by 3 to get -81. Write it under the 42. Add 42 + (-81) = -39.
Interpreting the Result
The numbers on the bottom row give you your answer. The very last number is the remainder (-39). The other numbers are the coefficients of your quotient polynomial. The quotient will always have a degree that is one less than the dividend. Since our dividend was a degree-3 polynomial, our quotient will be a degree-2 polynomial (a quadratic).
The coefficients are 1, -9, and -27. So, the quotient is x² - 9x - 27.
The Remainder Theorem and The Factor Theorem
Synthetic division is closely tied to two important theorems in algebra.
- The Remainder Theorem: This theorem states that if you divide a polynomial f(x) by (x - c), the remainder will be equal to f(c). In our example, the remainder was -39. If you were to plug x=3 into the original polynomial, f(3), you would also get -39. This provides a quick way to evaluate a function at a specific point.
- The Factor Theorem: This is a direct consequence of the Remainder Theorem. It states that (x - c) is a factor of a polynomial f(x) if and only if the remainder of the division is 0 (meaning f(c) = 0). This is the primary method used to test the possible rational roots of a polynomial.
Frequently Asked Questions
Why is it essential to include a zero for missing terms?
Each position in the synthetic division setup corresponds to a specific power of x (x³, x², x, constant). If you omit a term, you are essentially shifting all the subsequent terms to the left, which completely messes up the place value and will lead to an incorrect answer. Including a zero holds the place for the missing term.
What if my divisor is something like (2x - 6)?
Synthetic division only works for divisors in the form (x - c). However, you can often modify the problem. In the case of (2x - 6), you can factor out a 2 to get 2(x - 3). You would first perform synthetic division with c = 3. Then, you would divide the final quotient (but not the remainder) by the 2 that you factored out earlier.
How is this different from polynomial long division?
Synthetic division is a shortcut method that only works when the divisor is a linear binomial (x - c). Polynomial long division is a more general method that can be used to divide by a polynomial of any degree (e.g., a quadratic or cubic). Long division is more versatile, but synthetic division is much faster for the cases where it applies.