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How are algebraic expressions formed and manipulated?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — algebra This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

Algebraic expressions are formed by combining numbers and symbols—known as variables—using fundamental arithmetic operations such as addition, subtraction, and multiplication. In this system, letters like a, b, and c typically represent known constants, while letters like x, y, and z represent unknown variables. Exponential notation is also used to represent repeated multiplication, such as writing $x^{2}$ for $x \cdot x$ .12

These expressions are manipulated using established mathematical laws to simplify them or solve equations. Key principles include:

Structural Laws: The commutative, associative, and distributive laws are used to rearrange and simplify terms.13
Polynomial Operations: Polynomials can be added, subtracted, or multiplied to produce new polynomials.1
Rational Functions: By including division, expressions can be expanded into rational functions, which can be simplified by dividing the numerator by the denominator, similar to long division with integers.1

Today, these routine manipulations are often performed using symbolic algebra software, which automates the process of simplifying complex expressions.1