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Method

Synthetic Division

When dividing a polynomial by a linear binomial, a binomial of the form there is an alternative method to the polynomial long division called the synthetic division. This method uses the constant term of the binomial and the coefficients of the numerator to compute the quotient. Consider the following division.
This division can be found following the next six steps.
1
Set Up the Division Using and the Coefficients of the Numerator
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Recall that the denominator is a linear binomial in the form The denominator is so the value of is The division symbol for synthetic division is L-shaped. The number is written to the left.
After the numerator is written in standard form, its coefficients and constant term are written to the right of the division symbol. Fill in any missing terms with a zero.
The numerator does not have a linear term, so a was added between and Note that the number at the left of the division symbol is the opposite to the constant term of the divisor.
2
Bring Down the Leading Coefficient
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Bring the leftmost coefficient down across the line. In this case, is written below the horizontal line.
3
Multiply by the New Number Below the Line
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Multiply the number written under the line in the previous step by and write the product below the next coefficient above the line. In this case, is multiplied by The product is written in the next column below
4
Add the Numbers in the Column Above the Line
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Now, add the numbers in the column above the horizontal line, then write the result below the horizontal line in the same column. In this case, and are added and their sum, is written below them and under the horizontal line.
5
Repeat Steps and Through the Last Column
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Steps and are repeated through the last column. In this exercise, is multiplied by and the product is written in the next column and above the line.
The numbers in this column are added and the sum is written below the line.
Then is multiplied by and the product is written in the next column and above the line.
Finally, the numbers in the last column are added together and the sum is written below the line.
The division is now complete.
6
Write the Quotient and Remainder
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The rightmost number below the line is the remainder of the division. The remaining numbers below the line represent the coefficients of the quotient.
In this case, the remainder is The quotient is a quadratic polynomial with coefficients and
Notice that the quotient is always one degree lower than the numerator because the denominator is a linear binomial. For the same reason, the remainder is always a number.
The steps of synthetic division are shown below.
Division of x^3-3x^2+7 by x-2 using synthetic division
Once the quotient and the remainder are found, the original division can be written as the plus the division of the and the divisor.
Sometimes synthetic division is used to verify if a number is a root of a polynomial. This process is called synthetic substitution. A number is a root if the remainder of the division is
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