matha namo namaha


GUNÌTA SAMUCCAYAH - SAMUCCAYA GUŅÌTAH
In connection with factorization of quadratic expressions a sub-Sutra, viz. 'Gunita samuccayah-Samuccaya Gunitah' is useful. It is intended for the purpose of verifying the correctness of obtained answers in multiplications, divisions and factorizations. It means in this context:
'The product of the sum of the coefficients sc in the factors is equal to the sum of the coefficients sc in the product'
Symbolically we represent as sc of the product = product of the sc (in the factors)
Example 1:     (x + 3) (x + 2) = x2 + 5x + 6

        Now ( x + 3 ) ( x + 2 ) = 4 x 3 = 12 : Thus verified.
Example 2:     (x – 4) (2x + 5) = 2x2 – 3x – 20

        Sc of the product 2 – 3 – 20 = - 21

        Product of the Sc = (1 – 4) (2 + 5) = (-3) (7) = - 21. Hence verified.

        In case of cubics, biquadratics also the same rule applies.

        We have (x + 2) (x + 3) (x + 4) = x3 + 9x2 + 26x + 24

        Sc of the product = 1 + 9 + 26 + 24 = 60

        Product of the Sc = (1 + 2) (1 + 3) (1 + 4)

                                = 3 x 4 x 5 = 60. Verified.
Example 3:   (x + 5) (x + 7) (x – 2) = x3 + 10x2 + 11x – 70

                        (1 + 5) (1 + 7) (1 – 2) = 1 + 10 + 11 – 70

        i.e.,     6 x 8 x –1 = 22 – 70
        i.e.,     -48 = -48 Verified.
We apply and interpret So and Sc as sum of the coefficients of the odd powers and sum of the coefficients of the even powers and derive that So = Sc gives (x + 1) is a factor for thee concerned expression in the variable x. Sc = 0 gives (x - 1) is a factor.
Verify whether the following factorization of the expressions are correct or not by the Vedic check:
     i.e. Gunita. Samuccayah-Samuccaya Gunitah:

    1.     (2x + 3) (x – 2) = 2x2 – x - 6

    2.     12x2 – 23xy + 10y2 = ( 3x – 2y ) ( 4x – 5y )

    3.     12x2 + 13x – 4 = ( 3x – 4 ) ( 4x + 1 )

    4.     ( x + 1 ) ( x + 2 ) ( x + 3 ) = x3 + 6x2 + 11x + 6

    5.     ( x + 2 ) ( x + 3 ) ( x + 8 ) = x3 + 13x2 + 44x + 48
So far we have considered a majority of the upa-sutras as mentioned in the Vedic mathematics book. Only a few Upa-Sutras are not dealt under a separate heading . They are

    2) S’ISYATE S’ESASAMJ ÑAH
      
    4) KEVALAIH SAPTAKAMGUNYAT
      
    5) VESTANAM
      
    6) YAVADŨNAM TAVADŨNAM and

    10) SAMUCCAYAGUNITAH already find place in respective places.
        
Further in some other books developed on Vedic Mathematics DVANDAYOGA, SUDHA, DHVAJANKAM are also given as Sub-Sutras. They are mentioned in the Vedic Mathematics text also. But the list in the text (by the Editor) does not contain them. We shall also discuss them at appropriate places, with these three included, the total number of upa-Sutras comes to sixteen.
We now proceed to deal the Sutras with reference to their variety, applicability, speed, generality etc. Further we think how 'the element of choice in the Vedic system, even of innovation, together with mental approach, brings a new dimension to the study and practice of Mathematics. The variety and simplicity of the methods brings fun and amusement, the mental practice leads to a more agile, alert and intelligent mind and innovation naturally follow' (Prof. K.R.Williams, London).

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