Vedic Maths Sutra 15 : Guṇitasamuccayaḥ
Meaning & Essence
This Sūtra means “The product of the sum is equal to the sum of the products”.
In algebraic practice it is used as a powerful verification tool. When you multiply (or factor) polynomials, the sum of the coefficients of the product must equal the product of the sums of the coefficients of the factors. If the two sides match, the multiplication/factorization is correct.
Mathematical Principle
For two polynomials (or factors) \( P \) and \( Q \):
This is simply the evaluation of both sides at \( x = 1 \).
Worked Examples
- Claim: \( 8x^2 + 11x + 3 = (x + 1)(8x + 3) \)
- Sum of coefficients of the product: \( 8 + 11 + 3 = 22 \)
- Product of the sums of the factors: \( (1 + 1) \times (8 + 3) = 2 \times 11 = 22 \)
- Both sides equal → the factorization is correct
- \( (2 + 3)(4 + 5) = 5 \times 9 = 45 \)
- Expanded: \( 2\cdot4 + 2\cdot5 + 3\cdot4 + 3\cdot5 = 8 + 10 + 12 + 15 = 45 \)
- Both sides equal → correct
- Suppose someone claims \( (x + 2)(x + 5) = x^2 + 6x + 9 \)
- Sum of coefficients of claimed product: \( 1 + 6 + 9 = 16 \)
- Product of sums of factors: \( (1+2)(1+5) = 3 \times 6 = 18 \)
- 16 ≠ 18 → the expansion is wrong (correct is \( x^2 + 7x + 10 \))
Enter the coefficients of two linear factors and of the quadratic product.
The demo checks whether Guṇitasamuccayaḥ holds.
When to Use This Sūtra
• Verifying the correctness of polynomial multiplications
• Checking factorizations of quadratic (and higher) expressions
• Quickly detecting algebraic mistakes without expanding everything
• A simple but powerful “checksum” for algebraic work