## SYNTHETIC DIVISION

Synthetic division is a short cut method of polynomial division. The advantage of synthetic division is that it allows one to calculate without writing variables, than long division.

To use synthetic division in polynomial division, the divisor must of be of first degree and should be in the form

(x + a)  or  (x - a)

## Synthetic Division - Example

Find the quotient and remainder when

p(x)  =  3x3 - 2x2 - 5 + 7x

is divided by d(x)  =  x + 3 using synthetic division.

Solution :

Step 1 :

Arrange dividend and the divisor in standard form.

Then,

3x3 - 2x2 + 7x - 5     (standard form of dividend)

x + 3     (standard form of divisor)

Write the coefficients of dividend in the first row.

Put ‘0’ for missing term(s).

3     -2     7     -5     (first row)

Step 2 :

Find out the zero of the divisor.

x + 3  =  0

x  =  -3

Step 3 :

Write the zero of divisor in front of dividend in the first row.

Put ‘0’ in the first column of second row. Step 4 :

Complete the second row and third row as shown below. All the entries except the last one in the third row are the coefficients of the quotient.

Therefore, the quotient is

3x2 - 11x + 40

and the remainder is

-125

#### Division of Polynomials using Synthetic Division - Practice Questions

Question 1 :

Find the quotient and remainder using synthetic division :

(x3 + x2 - 3 x + 5) ÷ (x - 1)

Solution :

Arrange dividend and the divisor in standard form.

Then,

x3 + x2 - 3x + 5     (standard form of dividend)

x - 1     (standard form of divisor)

Find out the zero of the divisor.

x - 1  =  0

x  =  1 Therefore, the quotient is

x2 + 2x - 1

and the remainder is

4

Question 2 :

Find the quotient and remainder using synthetic division

(3x3 - 2x2 + 7x - 5) ÷ (x + 3)

Solution :

Arrange dividend and the divisor in standard form.

Then,

3x3 - 2x2 + 7x - 5     (standard form of dividend)

x + 3     (standard form of divisor)

Find out the zero of the divisor.

x + 3  =  0

x  =  -3 Therefore, the quotient is

3x2 - 11x + 40

and the remainder is

-125

Question 3 :

Find the quotient and remainder using synthetic division

(3x3 + 4x2 - 10x + 6) ÷ (3x - 2)

Solution :

Arrange dividend and the divisor in standard form.

Then,

3x3 + 4x2 - 10x + 6     (standard form of dividend)

3x - 2     (standard form of divisor)

Find out the zero of the divisor.

3x - 2  =  0

x  =  2/3 Therefore, the quotient is

3x2 + 6x - 6

and the remainder is

2 After having gone through the stuff given above, we hope that the students would have understood how to do polynomial division using synthetic division.

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