Gunitasamuchyah

Meaning: "The product of the sum (of coefficients in factors) is equal to the sum of the coefficients (in the product)". More simply, "Samuccaya (sum of coefficients) of the product of factors is equal to the Samuccaya of the product polynomial."

Primary Application: Checking Polynomial Multiplication and Factorization

This sutra serves as an excellent and quick verification tool in algebraic multiplication and factorization of polynomials. 'Samuccaya' in this context most often refers to the sum of the numerical coefficients of the terms in a polynomial. To get this sum, you can substitute the variable (e.g., x) with 1.

How it Works for Multiplication Verification:

1. Take two (or more) polynomials that are being multiplied.
2. For each polynomial factor, find the sum of its coefficients. (Let these sums be S₁, S₂, etc.)
3. Multiply these sums together: Product_of_Sums = S₁ x S₂ x ...
4. Now, look at the resulting product polynomial obtained after multiplication.
5. Find the sum of the coefficients of this product polynomial (Sum_of_Product_Coefficients).
6. Verification: If the multiplication is correct, then Product_of_Sums should be equal to Sum_of_Product_Coefficients.

How it Works for Factorization Verification:

1. You have a polynomial and its supposed factors.
2. Find the sum of coefficients of the original polynomial.
3. Find the sum of coefficients for each of the supposed factors.
4. Multiply the sums of coefficients of the factors.
5. Verification: If this product matches the sum of coefficients of the original polynomial, the factorization is likely correct (it's a necessary but not always sufficient condition, as different incorrect factorizations could coincidentally yield the same sum product).

Example: Check if (x+2)(x+3) = x² + 5x + 6.

• For (x+2): Coefficients are 1 (for x) and 2. Sum S₁ = 1+2 = 3.
• For (x+3): Coefficients are 1 (for x) and 3. Sum S₂ = 1+3 = 4.
• Product_of_Sums = S₁ x S₂ = 3 x 4 = 12.
• For the product polynomial x² + 5x + 6: Coefficients are 1, 5, and 6.
• Sum_of_Product_Coefficients = 1+5+6 = 12.

Since 12 = 12, the multiplication is verified by this sutra.

Note: This is related to the property that if P(x) = Q(x) * R(x), then P(1) = Q(1) * R(1), because P(1) is the sum of coefficients of P(x), and so on.