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Transpose and Apply is also known as " Parāvartya Yojayet".This method can be applied to solve Horner's process of Synthetic Division ( polynomial divisions ) and is useful in cases where the divisor consists of small digits.

Example :
Polynomial Divisions
x3+4x2+5x+1 by x-2
Here we have four terms of x3 , x2, x and an integer.We take each term separately and solve using four steps

Step 1 :
The x3 divided by x gives us x2 which is our first part of our result.
Result :
x2+......

Step 2:
x2 multiplied with -2 from (x-2)divider gives us the answer of -2x2.Now note that we have an integer of 4 in the above equation for x2.And so we add them together to get 6x2.Now divide this by x again ,we get answer 6x.
Result: 
x2 + 6x+.....

Step 3:
Now for the third term ,we take the result from step two and multiply it with the divider (-2) .We get the answer as 6x multiply -2 = -12x.We already have an co-efficient of   5 in the x -term and so adding them both ,we get 12+5 =17x.Now divide this by x again, we get answer 17.
Result :
x2 + 6x +17

Step 4:
Now the last term 17 from the above step  is multiplied by -2 .We get - 34 as answer.But we have 1 in the co-efficient and so adding them both we get 34+1 =35.This 35 is the Remainder.

Therefore for the polynomial equation  x3+4x2+5x+1 by  x-2 , we have
Quotient Q = x2 + 6x +17 and Remainder R =35

Note that the above method complements "All from nine and the last from ten", which is useful in divisions by large numbers.

This Sutra also has great use in dividing large Numbers by any number of digits .
Click here for its Continuation.


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