Algebra Cheat Sheet

The core ideas of Algebra distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Variables and Expressions

Variables are symbols, usually letters, that represent unknown or changeable quantities. Algebraic expressions combine variables, constants, and operations such as addition, subtraction, multiplication, and division into meaningful mathematical phrases.

Linear Equations

A linear equation is an equation in which the highest power of the variable is one. These equations graph as straight lines on the coordinate plane and have the general form $ax + b = c$. They are solved by isolating the variable using inverse operations.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, written in the standard form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula. Their graphs are parabolas that open upward or downward.

Systems of Equations

A system of equations is a set of two or more equations with the same variables that must be satisfied simultaneously. Common methods for solving systems include substitution, elimination, and graphing. A system can have one solution, no solution, or infinitely many solutions.

Polynomials

Polynomials are expressions consisting of variables raised to non-negative integer powers, multiplied by coefficients, and combined using addition or subtraction. The degree of a polynomial is the highest exponent of its variable. Operations on polynomials include addition, subtraction, multiplication, and division.

Factoring

Factoring is the process of breaking down a polynomial into simpler expressions (factors) whose product equals the original polynomial. Common techniques include factoring out the greatest common factor, grouping, and recognizing special patterns such as difference of squares or perfect square trinomials.

Functions

A function is a relation that assigns exactly one output to each input. Functions are described using notation like $f(x)$, where $x$ is the input and $f(x)$ is the output. They can be represented as equations, tables, graphs, or verbal descriptions and are fundamental to modeling real-world relationships.

Inequalities

Inequalities are mathematical statements that compare two expressions using symbols such as $<$, $>$, $\leq$, or $\geq$. Solving inequalities follows similar rules to solving equations, except that multiplying or dividing by a negative number reverses the inequality sign. Solutions are often expressed as intervals or graphed on a number line.

Exponents and Logarithms

Exponents represent repeated multiplication of a base by itself. Logarithms are the inverse operation of exponentiation, answering the question: to what power must a base be raised to produce a given number? These concepts are governed by rules such as the product rule, quotient rule, and power rule.

Matrices

Matrices are rectangular arrays of numbers arranged in rows and columns. They are used to represent and solve systems of linear equations, perform transformations in geometry, and organize data. Key operations include addition, scalar multiplication, matrix multiplication, and finding determinants and inverses.

Key Terms at a Glance

Absolute Value:The distance of a number from zero on the number line, always expressed as a non-negative value.

Binomial:A polynomial with exactly two terms, such as $3x + 5$ or $x^2 - 4$.

Coefficient:The numerical factor multiplied by a variable in an algebraic term. In $7x^2$, the coefficient is 7.

Constant:A fixed value that does not change. In the expression $4x + 9$, the number 9 is a constant.

Degree:The highest power of the variable in a polynomial. The degree of $3x^4 + x^2 - 1$ is 4.

Determinant:A scalar value computed from a square matrix that indicates whether the matrix is invertible and describes the scaling factor of the linear transformation.

Domain:The set of all permissible input values for a function.

Exponent:A number that indicates how many times a base is multiplied by itself. In $x^5$, the exponent is 5.

Expression:A mathematical phrase that combines numbers, variables, and operations but does not include an equals sign.

Factor:A number or expression that divides evenly into another. Factoring is the process of writing a quantity as a product of its factors.

Function:A relation in which each input value is paired with exactly one output value.

Inequality:A mathematical statement that compares two expressions using symbols such as $<$, $>$, $\leq$, or $\geq$.

Inverse:An operation or element that reverses the effect of another. The additive inverse of $a$ is $-a$; the multiplicative inverse of $a$ is $\frac{1}{a}$.

Linear Equation:An equation whose graph is a straight line and whose variable has a maximum exponent of one.

Logarithm:The inverse of exponentiation. $\log_b(x)$ equals the power to which the base $b$ must be raised to produce $x$.

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