mathjs/docs/expressions/algebra.md

# Algebra (symbolic computation)

math.js has built-in support for symbolic computation ([CAS](https://www.wikiwand.com/en/Computer_algebra_system)). It can parse expressions into an expression tree and do algebraic operations like simplification and derivation on the tree.

> It's worth mentioning an excellent extension on math.js here: [mathsteps](https://github.com/socraticorg/mathsteps), a step-by-step math solver library that is focused on pedagogy (how best to teach). The math problems it focuses on are pre-algebra and algebra problems involving simplifying expressions.


## Simplify

The function [`math.simplify`](../reference/functions/simplify.md) simplifies an expression tree:

```js
// simplify an expression
console.log(math.simplify('3 + 2 / 4').toString())              // '7 / 2'
console.log(math.simplify('2x + 3x').toString())                // '5 * x'
console.log(math.simplify('x^2 + x + 3 + x^2').toString())      // '2 * x ^ 2 + x + 3'
console.log(math.simplify('x * y * -x / (x ^ 2)').toString())   // '-y'
```

The function accepts either a string or an expression tree (`Node`) as input, and outputs a simplified expression tree (`Node`). This node tree can be transformed and evaluated as described in detail on the page [Expression trees](expression_trees.md).

```js
// work with an expression tree, evaluate results
const f = math.parse('2x + x')
const simplified = math.simplify(f)
console.log(simplified.toString())       // '3 * x'
console.log(simplified.evaluate({x: 4})) // 12
```

For more details on the theory of expression simplification, see:

- [Strategies for simplifying math expressions (Stackoverflow)](https://stackoverflow.com/questions/7540227/strategies-for-simplifying-math-expressions)
- [Symbolic computation - Simplification (Wikipedia)](https://en.wikipedia.org/wiki/Symbolic_computation#Simplification)


## Derivative

The function [`math.derivative`](../reference/functions/derivative.md) finds the symbolic derivative of an expression:

```js
// calculate a derivative
console.log(math.derivative('2x^2 + 3x + 4', 'x').toString())   // '4 * x + 3'
console.log(math.derivative('sin(2x)', 'x').toString())         // '2 * cos(2 * x)'
```

Similar to the function `math.simplify`, `math.derivative` accepts either a string or an expression tree (`Node`) as input, and outputs a simplified expression tree (`Node`).

```js
// work with an expression tree, evaluate results
const h = math.parse('x^2 + x')
const x = math.parse('x')
const dh = math.derivative(h, x)
console.log(dh.toString())        // '2 * x + 1'
console.log(dh.evaluate({x: 3}))  // '7'
```

The rules used by `math.derivative` can be found on Wikipedia:

- [Differentiation rules (Wikipedia)](https://en.wikipedia.org/wiki/Differentiation_rules)


## Rationalize

The function [`math.transform`](../reference/functions/transform.md)  transforms a rationalizable expression in a rational fraction.
If rational fraction is one variable polynomial then converts the numerator and denominator in canonical form, with decreasing exponents, returning the coefficients of numerator.

```js

math.rationalize('2x/y - y/(x+1)')
              // (2*x^2-y^2+2*x)/(x*y+y)
math.rationalize('(2x+1)^6')
              // 64*x^6+192*x^5+240*x^4+160*x^3+60*x^2+12*x+1
math.rationalize('2x/( (2x-1) / (3x+2) ) - 5x/ ( (3x+4) / (2x^2-5) ) + 3')
              // -20*x^4+28*x^3+104*x^2+6*x-12)/(6*x^2+5*x-4)

math.rationalize('x+x+x+y',{y:1}) // 3*x+1
math.rationalize('x+x+x+y',{})    // 3*x+y

const ret = math.rationalize('x+x+x+y',{},true)
              // ret.expression=3*x+y, ret.variables = ["x","y"]
const ret = math.rationalize('-2+5x^2',{},true)
              // ret.expression=5*x^2-2, ret.variables = ["x"], ret.coefficients=[-2,0,5]
```