# Putting Expressions into Different Forms

Complicated algebraic expressions can usually be written in many different ways. The Wolfram Language provides a variety of functions for converting expressions from one form to another.

In many applications, the most common of these functions are Expand, Factor, and Simplify. However, particularly when you have rational expressions that contain quotients, you may need to use other functions.

Expand[expr] | multiply out products and powers |

ExpandAll[expr] | apply Expand everywhere |

Factor[expr] | reduce to a product of factors |

Together[expr] | put all terms over a common denominator |

Apart[expr] | separate into terms with simple denominators |

Cancel[expr] | cancel common factors between numerators and denominators |

Simplify[expr] | try a sequence of algebraic transformations and give the smallest form of expr found |

Functions for transforming algebraic expressions.

Getting expressions into the form you want is something of an art. In most cases, it is best simply to experiment, trying different transformations until you get what you want. Often you will be able to use palettes in the front end to do this.

When you have an expression with a single variable, you can choose to write it as a sum of terms, a product, and so on. If you have an expression with several variables, there is an even wider selection of possible forms. You can, for example, choose to group terms in the expression so that one or another of the variables is "dominant".

Collect[expr,x] | group together powers of x |

FactorTerms[expr,x] | pull out factors that do not depend on x |

Rearranging expressions in several variables.

As you have seen, even when you restrict yourself to polynomials and rational expressions, there are many different ways to write any particular expression. If you consider more complicated expressions, involving, for example, higher mathematical functions, the variety of possible forms becomes still greater. As a result, it is totally infeasible to have a specific function built into the Wolfram Language to produce each possible form. Rather, the Wolfram Language allows you to construct arbitrary sets of transformation rules for converting between different forms. Many Wolfram Language packages include such rules; the details of how to construct them for yourself are given in "Transformation Rules and Definitions".

There are nevertheless a few additional built‐in Wolfram Language functions for transforming expressions.

TrigExpand[expr] | expand out trigonometric expressions into a sum of terms |

TrigFactor[expr] | factor trigonometric expressions into products of terms |

TrigReduce[expr] | reduce trigonometric expressions using multiple angles |

TrigToExp[expr] | convert trigonometric functions to exponentials |

ExpToTrig[expr] | convert exponentials to trigonometric functions |

FunctionExpand[expr] | expand out special and other functions |

ComplexExpand[expr] | perform expansions assuming that all variables are real |

PowerExpand[expr] | transform (xy)^{p} into x^{p}y^{p}, etc. |

Some other functions for transforming expressions.

The transformations on expressions done by functions like Expand and Factor are always correct, whatever values the symbolic variables in the expressions may have. Sometimes, however, it is useful to perform transformations that are only correct for some possible values of symbolic variables. One such transformation is performed by PowerExpand.