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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