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उपसूत्र १३. गुणितसमुच्चयः समुच्चयगुणितः

(Upasūtra 13. guṇitasmuccayaḥ samuccayaguṇitaḥ) - The product of the sum, is the sum of the products.
 

The Upasūtra: guṇitasmuccayaḥ samuccayaguṇitaḥ (The product of the sum, is the sum of the products) is used to verify factors of quadratic, cubic and even bi-quadratic expressions. It observes that the sum of the co-efficients of the factors, will be the same as the sum of the co-efficients in the product.
 
As an illustration, let us use this Upasūtra to verify:
6x2 + 13x + 5 = (2x + 1)(3x + 5)
  Steps 6x2 + 13x + 5 = (2x + 1)(3x + 5)
1. Write the expression in equated form, with the factors in one side, preferable LHS, and the product on the other, preferably RHS. In this case,
(2x + 1)(3x + 5) = 6x2 + 13x + 5
2. Find the product of the (Sum of all the co-efficients) of the factors. In this case,
LHS = (2 + 1) × (3 + 5) = 3 × 8
= 24
3. Find the sum of all the co-efficients of the product expression. In this case,
RHS = 6 + 13 + 5
= 24
4. These two sums will always be the same. If, not the factors are incorrect and the expression cannot be equated. In this case, since LHS = RHS, the expression in equated!
So, for a practitioner of Vedic Mathematics, for something like:
(2x + 3)(x - 2) = 2x2 - 5x - 6
  The above expression is not equated because:
Co-efficients of LHS = (2 + 3)(1 - 2) = -5
Co-efficients of RHS = 2 - 5 - 6 = -9
 
Thus, (2x + 3)(x - 2) 2x2 - 5x - 6
Again, for something like:
(x + 2)(x + 3)(x + 8) = x3 + 13x2 + 44x + 48
  Co-efficients of LHS = (1 + 2)(1 + 3)(1 + 8) = 108
Co-efficients of RHS = 1 + 13 + 44 + 48 = 106
 
Thus, (x + 2)(x + 3)(x + 8) x3 + 13x2 + 44x + 48
And, for something like:
(x + 1)(x + 2)(x + 3) = x3 + 6x2 + 11x + 6
  Co-efficients of LHS = (1 + 1)(1 + 2)(1 + 3) = 24
Co-efficients of RHS = 1 + 6 + 11 + 6 = 24
 
Thus, (x + 1)(x + 2)(x + 3) = x3 + 6x2 + 11x + 6, is equated.
Note that this method is used for verification, by nullifying the 'not equated' results, because as in the example above,
(x + 1)(x + 2)(x + 3) = x3 + 5x2 + 12x + 6
The Upasūtra would confirm that the expression above is equated, but it is not. As a result, this Upasūtra is used only for negating the non-equated options.
 
But, why does it work? For this Upasūtra (guṇitasmuccayaḥ samuccayaguṇitaḥ), let us consider the following:
  For any Quadratic expression:
(ax + b)(cx + d)
= ac x2 + bc x + ad x + bd
 
Considering only the co-efficients:
ac + bc + ad + bd
= a(c + d) + b(c + d)
= (a + b)(c + d)
 
Note, that Cubic and Bi-Quadratic expressions may be proven similarly.
 
Lastly, please remember that, as in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario - a guideline of which, is presented in Applications »
 
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