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Algebra

Build fluency with algebraic expressions, equations, functions, graphs, and polynomials.

For: Learners at different starting points, from those reviewing arithmetic foundations to those preparing for later mathematics.

After this page you can: Choose a suitable learning route and simplify expressions, solve equations, interpret functions, and work with polynomials.

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Part of: Mathematics

About Algebra

Algebra uses letters and symbols to represent quantities, express relationships, and work with unknown values. It includes simplifying expressions, solving equations and inequalities, describing patterns with functions, and representing relationships with graphs. These ideas connect: an expression can become part of an equation, and an equation can describe a function or a model.

Study algebra in steps that build on what you already know: review number operations as needed, learn how variables and expressions work, then practice solving equations and interpreting functions. Work through examples, check solutions in the original problem, and revisit a step when it is unclear. You can enter through the foundations, focus on equations, or choose a more specific goal such as polynomials and factoring.

Start here

route

I’m new to algebra

Begin with foundational language and skills, then choose a next step.

Start with: Elementary Algebra: Preface

route

I’m reviewing equations

Refresh the balance principle and practice solving equations one step at a time.

Start with: Solving Equations and Inequalities

Learn the foundations

1.Expressions and variablesVariables stand for numbers that can change, while expressions combine numbers and variables with operation symbols; simplifying includes using order of operat…

Variables stand for numbers that can change, while expressions combine numbers and variables with operation symbols; simplifying includes using order of operations and combining like terms.

  • Polynomials

    Continue from basic expressions to polynomial terms, degree, and operations.

Variable

A letter that represents a number whose value may change.

Expression

A number, a variable, or a combination of numbers and variables using operation symbols.

Try it

Simplify 3x + 2x.

5x, since the terms have the same variable raised to the same power.

2.Equations and inequalitiesSolving a linear equation uses properties of equality to keep both sides balanced, while solving an inequality also requires attention to its direction when mu…

Solving a linear equation uses properties of equality to keep both sides balanced, while solving an inequality also requires attention to its direction when multiplying or dividing by a negative number.

Equation

Two expressions connected by an equal sign.

Solution of an equation

A value of a variable that makes a true statement when substituted into the equation.

Try it

Solve x + 4 = 9.

x = 5; subtracting 4 from both sides preserves equality.

3.Functions and graphsA function assigns exactly one output to each input in its domain, and its relationship can be represented with notation, a table, an equation, or a graph.

A function assigns exactly one output to each input in its domain, and its relationship can be represented with notation, a table, an equation, or a graph.

Function

A relation in which each input in the domain leads to exactly one output.

Domain

The set of allowed input values for a function.

Try it

For f(x) = 2x + 1, find f(3).

f(3) = 7, found by replacing x with 3 and evaluating.

4.Polynomials and factoringPolynomials combine terms whose variable powers are nonnegative integers; factoring rewrites a polynomial as a product and can help solve some equations.

Polynomials combine terms whose variable powers are nonnegative integers; factoring rewrites a polynomial as a product and can help solve some equations.

  • Polynomials

    Review polynomial terms, degree, leading coefficient, and operations.

Polynomial

An expression made from a finite sum of terms with variables raised to nonnegative integer powers.

Degree

The highest power of the variable in a polynomial.

Try it

Factor x² + 5x + 6.

(x + 2)(x + 3), since the numbers 2 and 3 multiply to 6 and add to 5.

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Books

Glossary

Every free Algebra course and book on RedLectern

Free Algebra online courses

  1. A Vision of Linear AlgebraMIT RES.18-010 · Browse materials · Video
  2. Algebra IMIT 18.701 · Browse materials
  3. Algebra IIMIT 18.702 · Browse materials
  4. Calculus Revisited: Complex Variables, Differential Equations, and Linear AlgebraMIT RES.18-008 · Browse materials · Video
  5. Commutative AlgebraMIT 18.705 · Browse materials
  6. Engineering Math: Differential Equations and Linear AlgebraMIT 2.087 · Browse materials · Video
  7. Linear AlgebraMIT 18.06 · Browse materials · Video
  8. Linear AlgebraMIT 18.06SC · Browse materials · Video
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  1. Linear AlgebraMIT 18.700 · Browse materials
  2. Linear Algebra 1 (English)TU Delft OpenCourseWare · Browse materials · Video
  3. Modern AlgebraMIT 18.703 · Browse materials
  4. Noncommutative AlgebraMIT 18.706 · Browse materials
  5. Seminar in Algebra and Number Theory: Computational Commutative Algebra and Algebraic GeometryMIT 18.704 · Browse materials
  6. Seminar in Algebra and Number Theory: Rational Points on Elliptic CurvesMIT 18.704 · Browse materials
  7. Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of VariationsMIT 2.035 · Browse materials
  8. Algebra I Student NotesMIT RES.18-011 · Open on OCW
  9. Algebra II Student NotesMIT RES.18-012 · Open on OCW
  10. Applied Geometric AlgebraMIT RES.8-001 · Open on OCW
  11. Linear AlgebraMIT 18.06SC+summer_2019 · Open on OCW
  12. Linear Algebra - Communications IntensiveMIT 18.06CI · Open on OCW
How these courses are chosen

Listed as Algebra courses:

  • those whose title contains “Algebra”, or
  • those tagged “Algebra” as one of at most 2 topic tags, or as the first of 3.

Free Algebra textbooks and books

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