The Vedic Mathematics Sutra or word formula “Transpose and Apply” has a much wider application than the “transposition” which we know in Algebra as a short cut of the “golden rule’ of Algebra: do unto one side of the equation, what you do on the other side.
We first introduced this Sutra in MSC #12, Base Division. We will now apply this technique in polynomial division where the divisor is a quadratic equation.
In our example, the divisor is x2 + 2x + 3 and we transpose (reverse the signs of) the coefficients of all the terms of the divisor after the leading term. We also separate by a remainder bar the last two terms of the dividend. We write the transposed figures, – 2 and -3, below the original coefficients.
We obtain the first term of the quotient by dividing the first term of the dividend by the first term of the divisor (the First by the First), 2x3/ x2 = 2x. We then multiply the transposed figures by the coefficient of the first quotient term, 2, to get – 4 and -6, and place it under the corresponding terms of the dividend as shown in the illustration.
By adding the coefficient of the x2 term of the dividend 5 and -4, we get 1 the final figure in the answer. This is actually x2 but we have to divide it by x2 to get 1. Since the last answer figure is 1, we need only to place the transposed figures in the third row as shown.
The first figure in the remainder is the sum of 12x, – 6 and – 2 which is 4x, while the last figure is 12 + – 3 = 9.
Let’s take another example:
(4x3 + 12x2 + 8x – 5) ÷ (2x2 + 3x – 1) =
2x2 + 3x – 1) 4x3 + 12x2 | + 8x – 5
-3 1 – 6 2
. – 9 3
2x + 3 rem x – 2
Transpose the coefficients of the terms after the first term of the divisor and also place the remainder bar before the last two terms of the dividend.
Divide the first terms of the dividend by the last term of the divisor to get the first quotient term 4x3/ 2x2 = 2x. Multiply the transposed digits by 2 and place it in the row below the dividend. Deduct 6 from the coefficient of 12x2 to get 6x2 . divide this by 2x2 to get 3, the last quotient term.
Multiply the transposed digits by 3 to get – 9 and 3. The remainder is obtained by adding 8x + 2(x) – 9(x) = x and -5 + 3 = -2
More examples are discussed in Chapter 27 in Vol 3 of “30 Master Strategies in Computing”.