SYNTHETIC DIVISION

Synthetic division is a short cut method of polynomial division. The condition to be used this method is, the divisor must of be of first degree and should be in the form (x-a)

Let us learn the this method through an example:

Divide the following polynomial using Synthetic division :

Question 1

Find the quotient and remainder using synthetic division

( x³ + x² - 3 x + 5 ) ÷ ( x - 1 ) 

Solution :

Let p (x) = x³ + x² - 3 x + 5 be the dividend and q (x) = x - 1 be the divisor. We shall find the quotient s(x) and the remainder r, by proceeding as follows.

q (x) = 0

x - 1 = 0

     x = 1

Step 1 :

Arrange the dividend and the divisor according to the descending powers of x and then write the coefficients of dividend in the first zero. Insert 0 for missing terms.

Step 2 :

Find out the zero of the divisor.

Step 3 :

Put 0 for the first entry in the second row. 

Step 4 :

Write down the quotient and remainder accordingly. All the entries except the last one  in the third row constitute the coefficients of the quotient. 

When P (x) is divided by (x - 1), the quotient is x² + 2 x - 1 and the remainder is 4.

Quotient = x² + 2 x - 1

Remainder = 4

Question 2 :

Find the quotient and remainder using synthetic division

(3 x³ - 2 x² + 7 x - 5) ÷ (x + 3)

Solution :

Let p (x) = 3 x³ - 2 x² + 7 x - 5 be the dividend and q (x) = x + 3 be the divisor. We shall find the quotient s(x) and the remainder r, by proceeding as follows.

q (x) = 0

x + 3 = 0

     x = - 3

When P (x) is divided by (x + 3), the quotient is 3 x² - 11 x + 40 and the remainder is - 125.

Quotient = 3 x² - 11 x + 40

Remainder = - 125

Question 3 :

Find the quotient and remainder using synthetic division

(3 x³ + 4 x² - 10 x + 6) ÷ (3 x - 2)

Solution :

Let p (x) = 3 x³ + 4 x² - 10 x + 6 be the dividend and q (x) = 3 x - 2 be the divisor. We shall find the quotient s(x) and the remainder r, by proceeding as follows.

q (x) = 0

3 x - 2 = 0

    3 x = 2

       x = 2/3

When P (x) is divided by (x - 1), the quotient is 3 x² + 6 x - 6 and the remainder is 2.

3 x² + 6 x - 6

Dividing the whole equation by 3,we get

Quotient = x² + 2 x - 2

Remainder = 4

After having gone through the stuff given above, we hope that the students would have understood "Synthetic division"

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