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गुणकसमुच्चयः

Guṇakasamuccayaḥ

"The factors of the sum is the sum of the factors"

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What this sutra solves

Confirm a factorization is correct without expanding it out.

You claim (x+2)(x+3) factors x² + 5x + 6 and want a fast confidence check.

becomesIs the factorization right?
⚡

Test x = 1: factor side = 12, polynomial side = 12. They agree, so it holds.

Live Demo
Did we factor correctly?

Guṇakasamuccayaḥ: "the product of the sums equals the sum of the products." A true factorization is an identity — so it must hold for ANY value of x.

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Did we factor correctly?

Guṇakasamuccayaḥ: "the product of the sums equals the sum of the products." A true factorization is an identity — so it must hold for ANY value of x.

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⚡ Speed Advantage

Vedic
5 steps
Traditional
6 steps

See how few steps Vedic method needs!

Best for

  • • Verifying polynomial factorizations

Use when

  • • After factoring a polynomial — verify the result

Avoid when

  • • Primary computation

Intuition

The factors of an expression when evaluated at a value equal the expression's value — use this to verify factorizations.

Story Mode

The Factor Test

A factorization claims a polynomial splits into pieces. The ancient test: evaluate both the original and the product of factors at any point. They must agree. This is the digital fingerprint principle applied to algebra — a one-step sanity check on any factored form.

Vedic vs conventional

Conventional

multiply out factors and compare — many steps.

Vedic

evaluate at a test point — 1 step per factor.

Instant factorization verification vs. polynomial expansion.

Applications

Verifying polynomial factorizations

Confirm that a factorization is correct by evaluating at test values.

Verify (x+2)(x+3)=x2+5x+6(x+2)(x+3) = x^2+5x+6 at x=1x=1Verify (2x−1)(3x+2)=6x2+x−2(2x-1)(3x+2) = 6x^2+x-2 at x=0x=0 and x=1x=1

Common Mistakes to Avoid

✕

Verifying at only a convenient root

Wrong approach

Checking a factorization only at x=0x=0, where both sides happen to match.

Correct approach

✓
Use at least one meaningful test value, and for higher confidence test two simple values.

Why this happens

💡 Students choose the easiest substitution without asking whether it actually tests the factor structure.

✕

Treating a passed test as a complete proof

Wrong approach

If both sides match at x=1x=1, assuming the factorization must be fully correct.

Correct approach

✓
A test value is a fast verification aid. For a proof, expand or compare enough coefficients.

Why this happens

💡 The verification shortcut is mistaken for a full symbolic identity proof.

Why It Works

A claimed factorization is an identity:

P(x)=F(x)G(x)P(x)=F(x)G(x)

Evaluate both sides at a simple test value:

P(t)=?F(t)G(t)P(t)\stackrel{?}{=}F(t)G(t)

A mismatch disproves the factorization immediately:

P(t)≠F(t)G(t)⇒factorization is wrongP(t)\ne F(t)G(t)\Rightarrow \text{factorization is wrong}

This is a fast error check. Full proof still requires expansion or enough coefficient comparison.

If P(x)=(x−a)(x−b)P(x) = (x-a)(x-b), then P(a)=0P(a)=0 and P(b)=0P(b)=0. Evaluating the factors and the original polynomial at any value provides a consistency check on the factorization.

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Related Sutra

Guṇitasamuccayaḥ

The product of the sum is the sum of the products