Dividing Polynomials using Synthetic Division

What is synthetic division
Synthetic division is a shorthand method of polynomial division that’s particularly useful when dividing a polynomial by a linear factor of the form (x - c). It simplifies the long division process.

The following diagrams show how to divide polynomials using synthetic division. Scroll down the page for more examples and solutions.

 

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Steps for Synthetic Division

1. Write the coefficients of the dividend (the polynomial being divided) in descending order of their exponents. If any terms are missing, use a zero as a placeholder.
   
2. Write the value ‘c’ from the divisor (x - c) to the left of the coefficients.
   
3. Bring down the first coefficient of the dividend below the line.
   
4. Multiply and Add:
   Multiply the number you just brought down by ‘c’.
   Write the result below the next coefficient.
   Add the two numbers in that column.
   Write the sum below the line.
   
5. Repeat step 4 until you’ve reached the last coefficient.
   
6. Write the Result:
   The numbers below the line represent the coefficients of the quotient, starting with a degree one less than the dividend.
   The last number is the remainder.

Important Notes:

1. Synthetic division only works when dividing by a linear factor of the form (x - c).
   
2. Remember to use placeholders (zeros) for missing terms in the dividend.
   
3. If the remainder is not zero, you can express the result as:
   Quotient + (Remainder / Divisor)
   
4. Notice that you use the opposite sign of the constant term in the divisor (x - c). If the divisor is (x + 2), you use c = -2.
   
5. Remainder Theorem: The remainder when a polynomial f(x) is divided by (x - c) is equal to f(c).
   

How to divide polynomials using synthetic division?
This video shows how you can use synthetic division to divide a polynomial by a linear expression. It also shows how synthetic division can be used to evaluate polynomials.

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Synthetic Division
Another video showing how to use synthetic division to divide a polynomial by a linear expression and also how to use the remainder to evaluate the polynomial.

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Polynomial Division
Synthetic Division

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Algebra Homework Help - Synthetic Division
Here is a homework problem involving synthetic division. It is similar to polynomial division using long division but can be a quicker and easier way.

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