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fix: fixed #180 
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‎POLYNOMIAL_FACTORING.md‎

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# Polynomial Factoring Implementation (#33)
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This document summarizes the polynomial factoring implementation for issue #33 and issue #180.
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## Overview
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Implemented polynomial factoring algorithms to enable simplification of expressions like `sqrt(x²+2x+1)` → `|x+1|`.
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## Features Implemented
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### 1. Perfect Square Trinomial Factoring
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Detects and factors patterns like:
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- `a² + 2ab + b²` → `(a+b)²`
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- `a² - 2ab + b²` → `(a-b)²`
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Examples:
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```typescript
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sqrt(x² + 2x + 1) → |x+1|
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sqrt(x² - 2x + 1) → |x-1|
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sqrt(4x² + 12x + 9) → |2x+3|
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sqrt(a² + 2ab + b²) → |a+b|
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```
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### 2. Difference of Squares Factoring
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Detects and factors patterns like:
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- `a² - b²` → `(a-b)(a+b)`
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Examples:
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```typescript
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x² - 4 → (x-2)(x+2)
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4x² - 9 → (2x-3)(2x+3)
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```
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### 3. Quadratic Factoring with Rational Roots
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Factors quadratics using the quadratic formula when roots are rational:
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- `ax² + bx + c` → `a(x - r₁)(x - r₂)`
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Examples:
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```typescript
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x² + 5x + 6 → (x+2)(x+3)
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x² - 5x + 6 → (x-2)(x-3)
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```
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Note: Quadratics with irrational roots (like `x² + 2x - 1`) are not factored.
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## Integration with sqrt Simplification
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The factoring is automatically applied when simplifying square root expressions. Before applying the `sqrt(x²) → |x|` rule, the system now tries to factor the argument:
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```typescript
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// Before factoring implementation:
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sqrt(x² + 2x + 1) → sqrt(x² + 2x + 1) // No simplification
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// After factoring implementation:
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sqrt(x² + 2x + 1) → |x+1| // Factors to (x+1)², then applies sqrt rule
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```
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This fixes issue #180.
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## API
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### Exported Functions
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```typescript
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import {
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factorPerfectSquare,
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factorDifferenceOfSquares,
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factorQuadratic,
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factorPolynomial,
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} from '@cortex-js/compute-engine';
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// Factor perfect square trinomials
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factorPerfectSquare(expr: BoxedExpression): BoxedExpression | null
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// Factor difference of squares
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factorDifferenceOfSquares(expr: BoxedExpression): BoxedExpression | null
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// Factor quadratics with rational roots
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factorQuadratic(expr: BoxedExpression, variable: string): BoxedExpression | null
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// General polynomial factoring (tries all strategies)
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factorPolynomial(expr: BoxedExpression, variable?: string): BoxedExpression
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```
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### Usage Examples
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```typescript
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const ce = new ComputeEngine();
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// Automatic factoring in sqrt simplification
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const expr1 = ce.parse('\\sqrt{x^2 + 2x + 1}').simplify();
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console.log(expr1.latex); // \vert x+1\vert
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// Manual factoring
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import { factorPerfectSquare } from '@cortex-js/compute-engine';
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const expr2 = ce.parse('x^2 + 2x + 1');
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const factored = factorPerfectSquare(expr2);
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console.log(factored?.latex); // (x+1)^2
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```
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## Implementation Details
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### Files Modified
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1. **src/compute-engine/boxed-expression/factor.ts**
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- Added `factorPerfectSquare()` - detects perfect square trinomials
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- Added `factorDifferenceOfSquares()` - detects difference of squares
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- Added `factorQuadratic()` - factors using quadratic formula for rational roots
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- Added `factorPolynomial()` - general factoring dispatcher
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- Added `extractSquareRoot()` - helper to extract square roots safely
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2. **src/compute-engine/symbolic/simplify-power.ts**
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- Integrated factoring into sqrt simplification
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- Added perfect square and difference of squares checks before applying sqrt rules
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3. **src/compute-engine/index.ts**
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- Exported new factoring functions for public API
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### Key Implementation Considerations
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1. **No .simplify() calls in factoring functions**
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- To avoid infinite recursion, factoring functions don't call `.simplify()` on their results
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- However, `.simplify()` is called on individual square roots during extraction, which is safe
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- See CLAUDE.md section "Simplification and Recursion Prevention" for details
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2. **Handling different representations**
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- `4x²` is represented as `Multiply(4, Power(x, 2))`, not `Power(2x, 2)`
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- The implementation uses `.sqrt().simplify()` to extract square roots properly
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- Handles both symbolic (√x²) and numeric (√4) perfect squares
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3. **Irrational root filtering**
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- Quadratic factoring checks for radical components in discriminant and roots
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- Uses `numericValue.radical` property to detect irrational square roots
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- Only factors when roots are provably rational
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## Tests
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Comprehensive test suite in `test/compute-engine/factor.test.ts`:
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- Perfect square trinomial tests (6 tests)
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- Difference of squares tests (6 tests)
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- Quadratic factoring tests (8 tests)
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- General polynomial factoring tests (4 tests)
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- Integration with sqrt simplification (8 tests)
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- Issue #180 regression tests (3 tests)
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All tests passing ✓
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## Performance
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The factoring algorithms are O(1) for checking patterns:
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- Perfect square: O(1) - checks 3 terms
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- Difference of squares: O(1) - checks 2 terms
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- Quadratic factoring: O(1) - quadratic formula
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The algorithms are called during simplification only when needed (sqrt of Add expressions).
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## Future Enhancements
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Potential improvements not yet implemented:
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1. Higher-degree polynomial factoring (cubic, quartic)
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2. Factoring over different domains (complex numbers, modular arithmetic)
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3. Multivariate polynomial factoring
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4. Kronecker's method for general factorization
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## Related Issues
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- Issue #180: "Factoring before trying to simplify" ✓ Fixed
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- Issue #33: "Polynomial Factoring" ✓ Implemented

‎examples/polynomial-factoring.js‎

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// Polynomial Factoring Examples
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// Demonstrates the new polynomial factoring capabilities
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import { ComputeEngine } from '../dist/compute-engine.esm.js';
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const ce = new ComputeEngine();
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console.log('='.repeat(60));
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console.log('POLYNOMIAL FACTORING EXAMPLES');
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console.log('='.repeat(60));
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console.log('\n1. Perfect Square Trinomials');
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console.log('-'.repeat(40));
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console.log('sqrt(x² + 2x + 1) =', ce.parse('\\sqrt{x^2 + 2x + 1}').simplify().latex);
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console.log('sqrt(x² - 2x + 1) =', ce.parse('\\sqrt{x^2 - 2x + 1}').simplify().latex);
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console.log('sqrt(a² + 2ab + b²) =', ce.parse('\\sqrt{a^2 + 2ab + b^2}').simplify().latex);
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console.log('sqrt(a² - 2ab + b²) =', ce.parse('\\sqrt{a^2 - 2ab + b^2}').simplify().latex);
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console.log('\n2. Perfect Squares with Coefficients');
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console.log('-'.repeat(40));
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console.log('sqrt(4x² + 12x + 9) =', ce.parse('\\sqrt{4x^2 + 12x + 9}').simplify().latex);
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console.log('sqrt(4x² - 12x + 9) =', ce.parse('\\sqrt{4x^2 - 12x + 9}').simplify().latex);
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console.log('sqrt(9x² + 6x + 1) =', ce.parse('\\sqrt{9x^2 + 6x + 1}').simplify().latex);
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console.log('\n3. Difference of Squares');
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console.log('-'.repeat(40));
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console.log('sqrt((x² - 4)(x² + 4)) =', ce.parse('\\sqrt{(x^2 - 4)(x^2 + 4)}').simplify().latex);
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console.log('sqrt((x - 2)(x + 2)) =', ce.parse('\\sqrt{(x - 2)(x + 2)}').simplify().latex);
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console.log('\n4. Non-Perfect Squares (unchanged)');
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console.log('-'.repeat(40));
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console.log('sqrt(x² + 3x + 1) =', ce.parse('\\sqrt{x^2 + 3x + 1}').simplify().latex);
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console.log('sqrt(x² + x + 1) =', ce.parse('\\sqrt{x^2 + x + 1}').simplify().latex);
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console.log('\n5. Issue #180 Examples');
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console.log('-'.repeat(40));
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console.log('Expanded form: sqrt(x² + 2x + 1) =', ce.parse('\\sqrt{x^2+2x+1}').simplify().latex);
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console.log('Factored form: sqrt((x+1)²) =', ce.parse('\\sqrt{(x+1)^2}').simplify().latex);
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console.log('Both now simplify to: |x+1|');
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console.log('\n6. Using the factorPolynomial Function');
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console.log('-'.repeat(40));
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import {
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factorPerfectSquare,
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factorDifferenceOfSquares,
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factorQuadratic,
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factorPolynomial,
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} from '../dist/compute-engine.esm.js';
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const expr1 = ce.parse('x^2 + 2x + 1');
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console.log('x² + 2x + 1 factored:', factorPerfectSquare(expr1)?.latex || 'null');
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const expr2 = ce.parse('x^2 - 4');
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console.log('x² - 4 factored:', factorDifferenceOfSquares(expr2)?.latex || 'null');
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const expr3 = ce.parse('x^2 + 5x + 6');
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console.log('x² + 5x + 6 factored:', factorQuadratic(expr3, 'x')?.latex || 'null');
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const expr4 = ce.parse('4x^2 + 12x + 9');
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console.log('4x² + 12x + 9 factored:', factorPolynomial(expr4)?.latex || 'null');
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console.log('\n' + '='.repeat(60));
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console.log('All examples completed successfully!');
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console.log('='.repeat(60));

‎examples/polynomial-factoring.ts‎

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// Polynomial Factoring Examples
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// Demonstrates the new polynomial factoring capabilities
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// Run with: npx tsx examples/polynomial-factoring.ts
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import { ComputeEngine } from '../src/compute-engine/index.ts';
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import {
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factorPerfectSquare,
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factorDifferenceOfSquares,
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factorQuadratic,
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factorPolynomial,
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} from '../src/compute-engine/boxed-expression/factor.ts';
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const ce = new ComputeEngine();
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console.log('='.repeat(60));
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console.log('POLYNOMIAL FACTORING EXAMPLES');
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console.log('='.repeat(60));
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console.log('\n1. Perfect Square Trinomials');
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console.log('-'.repeat(40));
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console.log('sqrt(x² + 2x + 1) =', ce.parse('\\sqrt{x^2 + 2x + 1}').simplify().latex);
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console.log('sqrt(x² - 2x + 1) =', ce.parse('\\sqrt{x^2 - 2x + 1}').simplify().latex);
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console.log('sqrt(a² + 2ab + b²) =', ce.parse('\\sqrt{a^2 + 2ab + b^2}').simplify().latex);
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console.log('sqrt(a² - 2ab + b²) =', ce.parse('\\sqrt{a^2 - 2ab + b^2}').simplify().latex);
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console.log('\n2. Perfect Squares with Coefficients');
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console.log('-'.repeat(40));
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console.log('sqrt(4x² + 12x + 9) =', ce.parse('\\sqrt{4x^2 + 12x + 9}').simplify().latex);
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console.log('sqrt(4x² - 12x + 9) =', ce.parse('\\sqrt{4x^2 - 12x + 9}').simplify().latex);
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console.log('sqrt(9x² + 6x + 1) =', ce.parse('\\sqrt{9x^2 + 6x + 1}').simplify().latex);
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console.log('\n3. Non-Perfect Squares (unchanged)');
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console.log('-'.repeat(40));
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console.log('sqrt(x² + 3x + 1) =', ce.parse('\\sqrt{x^2 + 3x + 1}').simplify().latex);
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console.log('sqrt(x² + x + 1) =', ce.parse('\\sqrt{x^2 + x + 1}').simplify().latex);
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console.log('\n4. Issue #180 Examples');
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console.log('-'.repeat(40));
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console.log('Expanded form: sqrt(x² + 2x + 1) =', ce.parse('\\sqrt{x^2+2x+1}').simplify().latex);
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console.log('Factored form: sqrt((x+1)²) =', ce.parse('\\sqrt{(x+1)^2}').simplify().latex);
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console.log('Both now simplify to: |x+1|');
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console.log('\n5. Using the factorPolynomial Function');
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console.log('-'.repeat(40));
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const expr1 = ce.parse('x^2 + 2x + 1');
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console.log('x² + 2x + 1 factored:', factorPerfectSquare(expr1)?.latex || 'null');
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const expr2 = ce.parse('x^2 - 4');
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console.log('x² - 4 factored:', factorDifferenceOfSquares(expr2)?.latex || 'null');
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const expr3 = ce.parse('x^2 + 5x + 6');
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console.log('x² + 5x + 6 factored:', factorQuadratic(expr3, 'x')?.latex || 'null');
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const expr4 = ce.parse('4x^2 + 12x + 9');
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console.log('4x² + 12x + 9 factored:', factorPolynomial(expr4)?.latex || 'null');
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console.log('\n' + '='.repeat(60));
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console.log('All examples completed successfully!');
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console.log('='.repeat(60));

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