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 expression.
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), 2x4/ x2 = 2x2.
We then multiply the coefficient of the first quotient term, 2, by the transposed figures to get 4 and -6, and place it under the 2nd and 3rd terms of the dividend as shown in the illustration.
By adding the coefficient of the x3 term of the dividend, -1, and 4, we get 3, the coefficient of the x term in the answer.
Again, we multiply this 3 by the transposed figures to give 6 and -9 which we will place in the 3rd and 4th columns of our solution.
We the add 1 which is the coefficient of the x2 term in the dividend, -6 and 6 to get 1 as the last answer figure. This is actually 1x2 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 fourth row as shown.
The first figure in the remainder is the sum of 9(x), – 9 and 2 which is 2x, while the last figure is 8 + – 3 = 5.
So we have (2x4 – x3 + x2 + 9x + 8) ÷ (x2 – 2x + 3) = 2x2 + 3x + 1 rem 2 x + 5