Practice
Algebra
Courses
Take a guided, problem-solving based approach to learning Algebra. These compilations provide unique perspectives and applications you won't find anywhere else.
Algebra Fundamentals
What's inside
- Introduction
- Simplifying Shortcuts
- Arithmetic Logic and Magic
- Balancing Scales
- Sequences
- Rates and Ratios
Algebra I
What's inside
- Introduction
- Equations and Unknowns
- Manipulating Exponents
- Algebra in Motion
- Factorials
- Common Misconceptions
Algebra II
What's inside
- Introduction
- Function Fundamentals
- Transformations
- Powers and Radicals
- Polynomials
- Factoring Polynomials
- Rational Functions
- Piecewise Functions
Additional Practice
Sharpen your skills with these quizzes designed to check your understanding of the fundamentals.
Expressions and Variables
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Algebra Warmups
From cracking cryptograms to calculating the top speed of a rocket, algebra gives you tools to apply mathematical reasoning to a wide range of problems. Dive in and see what you already know!
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Properties of Arithmetic
Understanding the properties of arithmetic will allow you to simplify complex expressions and do difficult calculations in a flash. Just don't divide by 0...
Solving Equations
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Solving Equations
From supply and demand to position and velocity, equations model the world around us. Equip yourself with the tools necessary to solve for x.
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Common Misconceptions (Algebra)
Arm yourself with the tools to be the king or queen of heady mathematical debates, like the age old question of whether 0.999.... = 1.
Linear Equations
Systems of Linear Equations
Linear Inequalities
Non-Linear Inequalities
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Polynomial Inequalities
When does revenue exceed cost? When will one race car pass another? Model these values with polynomials and use polynomial inequalities to solve questions like these and more.
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Absolute Value Inequalities
Is a company's profit greater in magnitude than its loss? When is the error in the length of a candy bar acceptable to the factory?
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Exponential Inequalities
When will a bacteria colony reach a certain size? When will one investment outperform another? Many real-world models use exponential functions, and you'll need these tools to compare them.
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Logarithmic Inequalities
Logarithmic inequalities are useful in analyzing situations involving repeated multiplication, such as interest and exponential decay.
Quadratic Equations (Parabolas)
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Quadratic Equations
Quadratic equations describe the motion of a baseball after it connects with a bat, and the acceleration of gravity at the Earth's orbit.
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Completing The Square
Learn how to eliminate the linear term and see through messy quadratics. You’ll know just how high to shoot your three-pointer to avoid tall defenders.
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Quadratic Discriminant
This one number can tell you whether the solutions to a quadratic equation are real or non-real, and whether they are distinct or repeated.
Square Roots (Radicals)
Arithmetic and Geometric Progressions
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Arithmetic Progressions
What's the sum of the first 100 positive integers? How about the first 1000? Learn the fun and fast way to solve problems like this.
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Geometric Progressions
A geometric progression is a sequence of numbers where the previous term is multiplied by a constant to get the next term. 1, 2, 4, 8,... is a geometric sequence where each term is multiplied by 2.
Functions
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Functions
Functions map an input to an output. For example, the function f(x) = 2x takes an input, x, and multiplies it by two. An input of x = 2 gives you an output of 4. Learn all about functions.
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Function Graphs
Graphs are visual representations of functions. If you know how to read graphs, you can say a lot about a function just by looking at its graph. Learn this fine art of mathematical divining.
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Functional Equations
Functional equations are equations where the unknowns are the functions. Rather than solving for x, you solve for the function in questions like "Find all functions that have these properties."
Polynomials
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Polynomial Arithmetic
From profit/cost models to the flight of a baseball to approximating the motion of a wave, polynomials are a mathematical way to represent values which are sums of powers of variables.
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Polynomial Factoring
A factored polynomial reveals its roots, a key concept in understanding the behavior of these expressions.
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Remainder Factor Theorem
You might have heard of remainders in simple division, but what about remainders when dividing polynomials?
Rational Functions
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Rational Functions
A rational function can have a variable like "x" in the numerator AND the denominator. When this happens, there are some special rules and properties to consider.
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Partial Fractions
Express rational functions as a sum of fractions with simpler denominators. You can apply this to telescoping series to mass cancel terms in a seemingly complicated sum.
Piecewise Functions
Exponents
Exponential Functions
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Exponential Functions
From compound interest to bubonic plague, things that grow or spread really fast are often modeled by exponential functions. Learn about these powerful functions (pun intended?).
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Exponential Inequalities
When will a bacteria colony reach a certain size? When will one investment outperform another? Many real-world models use exponential functions, and you'll need these tools to compare them.
Logarithms
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Logarithmic Functions
Logarithmic scales are used when the range of possible values is very wide, such as the intensity of an earthquake or the acidity of a liquid, in order to avoid the use of large numbers.
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Logarithmic Inequalities
Logarithmic inequalities are useful in analyzing situations involving repeated multiplication, such as interest and exponential decay.
Complex Numbers
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Complex Numbers
"Complex" numbers share many properties with the real numbers. In fact, complex numbers include all the real numbers, so you know lots of them already. Such an overachiever.
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De Moivre's Theorem
De Moivre's Theorem shows that to raise a complex number to the nth power, the absolute value is raised to the nth power and the argument is multiplied by n.
Advanced Linear Algebra (Matrices)
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Linear Algebra
Apply linear algebra to solve systems of linear equations, find paths in graph theory, and map rotations of points in space using matrix operations. It's a true intersection of engineering and math.
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Systems of Equations
Vieta root jumping is a descent method that occurs when you have to solve a Diophantine equation whose solutions have a recursive structure. Popular in advanced math olympiad number theory problems.
Advanced Polynomials
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Algebraic Manipulation
Gauss, Ramanujan, and a pantheon of other mathematicians have given us algebraic manipulation tools way beyond what's taught in school. Learn what they knew.
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Binomial Theorem
If a coin comes up heads you win $10, but if it comes up tails you win $0. How likely are you to win exactly $30 in five flips?
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Rational Root Theorem
The rational root theorem describes a relationship between the roots of a polynomial and its coefficients. For a good time, use the theorem to prove that the square root of 2 is not rational.
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Vieta's Formula
Vieta's formula relates the coefficients of polynomials to the sum and products of their roots. This can provide a shortcut to finding solutions in more complicated algebraic polynomials.
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Completing The Square Generalized
This is completing the square generalized to higher degree polynomials!
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Advanced Factorization
Advanced factorization is a gateway to algebraic number theory, which mathematicians study in order to solve famous conjectures like Fermat's Last Theorem.
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Polynomial Interpolation
Suppose you have a system of 50 linear equations. It's tedious and impractical to solve them by hand. Polynomial interpolation is your better alternative.
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Roots of Unity
A root of unity is a complex number that, when raised to a positive integer power, results in 1. Roots of unity have applications to the geometry of regular polygons, group theory, and number theory.
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Symmetric Polynomials
In a symmetric polynomial, you can interchange any of the variables and get the same polynomial. Symmetric polynomials form the basis of Galois theory, which connects group theory and field theory.
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Chebyshev Polynomials
Chebyshev polynomials are a sequence of orthogonal polynomials that provide recurrence relations useful for solving polynomials and approximating functions without extensive calculation.
Advanced Inequalities
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Classical Inequalities
Your destination for questions of the form "do we know if this expression is always greater than this other expression?" Explore what humans know about mathematical inequalities.
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Power Mean Inequalities
This chain of inequalities forms the foundation for many other classical inequalities. See how the four common "means" - arithmetic, geometric, harmonic, and quadratic - relate to each other.
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Geometric Inequalities
Given 5 sticks of length 1, 3, 5, 9, and 10, how many distinct triangles can be formed? Learn the techniques and develop an intuition for working with geometric inequalities.
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Math Competition Preparation
The smell of linoleum. Apple juice and cookies at halftime. And extra hard math problems. Do it for the love, the thrill. Learn more about getting in on the action of math competitions.
Community Wiki
Browse through thousands of Algebra wikis written by our community of experts.
Expressions and Variables
Solving Equations
- Setting Up Equations
- Simple Equations
- Solving Equations
- Verifying Solutions
- Multi-step Equations
- Isolating a Variable
- Balance Puzzles
- System of Linear Equations (Simultaneous Equations)
- How are exponent towers evaluated?
- Does a square root have two values?
- Do Square Roots Always Multiply?
- Is 0.999... = 1?
- What is 0 to the power of 0?
- If AB=AC, does B=C?
- Is (a/b)/c = a/(b/c)?
- Does x2+y2=x+y?
- Does cross multiply always work for inequalities?
- How does addition in the denominator work?
- List of Common Misconceptions
Linear Equations
Systems of Linear Equations
Absolute Value
Linear Inequalities
Non-Linear Inequalities
Quadratic Equations (Parabolas)
Square Roots (Radicals)
Arithmetic and Geometric Progressions
Functions
Polynomials
- Polynomials
- Multiplying Polynomials
- Polynomial Division
- Synthetic Division
- Solving Identity Equations
- Difference Of Squares
- Applying the Perfect Square Identity
- Applying the Perfect Cube Identity
- Factoring Polynomials
- Factoring by Substitution
- Rational Expressions
- Simplifying Rational Expressions
- Factoring Compound Quadratics: ax4+bx2+c
- Factoring Cubic Polynomials
- Descartes' Rule of Signs
- Fundamental Theorem of Algebra
- Method of Undetermined Coefficients
- Remainder Factor Theorem
- Transforming Roots of Polynomials
Rational Functions
Piecewise Functions
Exponents
Exponential Functions
Logarithms
Polar Coordinates
Parametric Equations
Complex Numbers
Advanced Linear Algebra (Matrices)
- Linear Algebra
- Matrices
- Determinants
- Eigenvalues and Eigenvectors
- Kernel (Nullspace)
- Vector Space
- Rank
- Cayley-Hamilton Theorem
- Row And Column Spaces
- Spectral Theorem
- Fundamental Subspaces
- Change of Basis
- Basis
- Rank-Nullity Theorem
- Linear Transformations
- Linear Independence
- Jordan Canonical Form
- Affine transformations
- Vieta Root Jumping
Advanced Polynomials
- Algebraic Manipulation
- Algebraic Manipulation Identities
- Sum of n, n², or n³
- Telescoping Series - Product
- Nested Functions
- Cardano's Method
- Cubic Discriminant
- Gauss: The Prince of Mathematics
- Srinivasa Ramanujan
- Binomial Coefficient
- Pascal's Triangle
- Binomial Theorem
- Negative Binomial Theorem
- Fractional Binomial Theorem
- Rational Root Theorem
- Vieta's Formula
- Newton's Identities
- Completing the Square - Multiple Variables
- Algebraic Identities
- Factorization of Polynomials
- Sophie Germain Identity
- Algebraic Number Theory
- Fermat's Last Theorem
- Eisenstein's Irreducibility Criterion
- Group Theory
- Lagrange's Theorem
- Schwartz-Zippel Lemma
- Polynomial Interpolation by Remainder Factor Theorem
- Lagrange Interpolation
- Method of Differences
- Primitive Roots of Unity
- Symmetric Polynomials
- The uvw Method
- Chebyshev Polynomials - Definition and Properties
Advanced Inequalities
- Cauchy-Schwarz Inequality
- Titu's Lemma
- Rearrangement Inequality
- Chebyshev's Inequality
- Jensen's Inequality
- Hölder's Inequality
- Young's Inequality
- Muirhead Inequality
- Reverse Rearrangement Inequality
- Arithmetic Mean - Geometric Mean
- Applying the Arithmetic Mean Geometric Mean Inequality
- Power Mean Inequality (QAGH)
- Quadratic Mean
- Triangle Inequality
- AIME Math Contest Preparation
- JEE Exam Preparation
- RMO Math Contest Preparation
- INMO Math Contest Preparation
- KVPY Exam Preparation
- BITSAT Exam Preparation