Algebra Terms

Expressions 13

• CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Grade 6, Grade 7, Grade 8
• Combining Like TermsCombining like terms is simplifying an expression by adding or subtracting the terms that have exactly the same variable part. You add their coefficients and keep the variable part: 3x + 2x = 5x. Constants combine with constants. So 3x + 5 + 2x − 1 simplifies to 5x + 4.Grade 6, Grade 7, Algebra 1
• ConstantA constant is a fixed value that does not change. In an expression, a constant is a number without a variable, such as 8 in x + 8 or −3 in 4a − 3. Special numbers such as π are constants too. Changing the variable changes the other terms, but never the constant.Grade 6, Grade 7, Grade 8
• Constant TermA constant term is a term in an expression or equation that has no variable, so its value never changes. In 3x^2 − 5x + 7, the constant term is 7. Its sign belongs to it: in x^2 + 4x − 9 it is −9. In a word problem it is usually a fixed amount, such as a one-time fee.Grade 6, Grade 7, Algebra 1
• Equivalent ExpressionsEquivalent expressions are expressions that have the same value for every value of the variable, even though they look different. For example, 3(x + 2) and 3x + 6 are equivalent: both equal 15 when x = 3 and 0 when x = −2. You build them with the properties of operations, such as the distributive property.Grade 6, Grade 7, Algebra 1
• EvaluateEvaluate means to find the single number value of an expression. Replace each variable with the number you are given, then calculate using the order of operations. For example, 2x + 7 evaluated at x = 3 is 2(3) + 7 = 13. The result is one number, not a new expression.Grade 6, Grade 7, Algebra 1
• ExpressionAn expression is a number, variable, or combination of numbers and variables joined by operations, without an equal sign. Examples are 7, 4x + 9 and 3(a − 2). Expressions can be simplified, but they are not solved unless they are part of an equation. You evaluate one by replacing each variable with a number.Grade 6, Grade 7, Grade 8
• Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a^2 and −2a^2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Grade 6, Grade 7, Grade 8
• Order of OperationsThe order of operations is the agreed sequence for evaluating an expression: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Without it the same expression would have several answers. Many students remember it as PEMDAS.Grade 5, Grade 6, Grade 7
• PEMDASPEMDAS is an acronym for remembering the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Multiplication and division share one step and are done from left to right, and so are addition and subtraction. Many students learn it as "Please Excuse My Dear Aunt Sally."Grade 5, Grade 6, Grade 7
• SubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Grade 6, Grade 7, Grade 8
• TermA term is a single number, a variable, or numbers and variables multiplied together, such as 7, x or −4xy^2. In an expression, terms are the parts separated by plus and minus signs, so 5x^2 − 3x + 8 has three terms. The sign in front of a term belongs to it.Grade 6, Grade 7, Algebra 1
• VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Grade 6, Grade 7, Grade 8

Properties 1

• Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Grade 6, Grade 7, Grade 8

Exponents 6

• BaseA base is the number or expression that is repeatedly multiplied when using an exponent. In 4^5 the base is 4 and the exponent is 5, so 4^5 = 4 × 4 × 4 × 4 × 4 = 1024. The base is the big number written on the line; the exponent is the small raised number.Grade 6, Grade 7, Grade 8
• ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 5^3, the base is 5 and the exponent is 3, so 5^3 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Grade 6, Grade 7, Grade 8
• Laws of ExponentsThe laws of exponents are the rules for simplifying powers that share a base or an exponent. Multiply powers of the same base by adding exponents, divide by subtracting them, and raise a power to a power by multiplying them: x^2 · x^5 = x^7, (x^6)/(x^2) = x^4, (x^3)^2 = x^6.Grade 8, Algebra 1, Algebra 2
• Negative ExponentA negative exponent tells you to take the reciprocal of the base raised to the matching positive exponent: a^(−n) = 1/(a^n) for a ≠ 0. So 2^(−3) = 1/8 and 10^(−2) = 0.01. A negative exponent makes a number smaller in size, but it never makes the value negative.Grade 8, Algebra 1
• Rational ExponentA rational exponent is an exponent written as a fraction, and it stands for a root. The denominator is the root and the numerator is the power: a^(1/n) = nth root of a and a^(m/n) = ( nth root of a )^m. For example, 9^(1/2) = 3 and 27^(2/3) = 3^2 = 9.Algebra 1, Algebra 2
• Zero ExponentA zero exponent is an exponent of 0, and any nonzero number raised to the power 0 equals 1. So 7^0 = 1, (−3)^0 = 1 and x^0 = 1 for x ≠ 0. The rule keeps the quotient rule working: (5^3)/(5^3) = 5^0, and a number divided by itself is 1.Grade 8, Algebra 1

Polynomials and factoring 12

• BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a^2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra 1, Algebra 2
• Degree of a PolynomialThe degree of a polynomial is the highest degree of any of its terms. For one variable, that is the largest exponent: 4x^3 − 2x^5 + 7x has degree 5. For a term with several variables, add their exponents, so 3x^2y^4 has degree 6. A nonzero constant such as 9 has degree 0.Algebra 1, Algebra 2
• Difference of SquaresA difference of squares is an expression of the form a^2 − b^2: one perfect square subtracted from another. It always factors as (a + b)(a − b). For example, x^2 − 49 = (x + 7)(x − 7). A sum of squares, a^2 + b^2, does not factor over the real numbers.Algebra 1, Algebra 2
• Factor CompletelyFactor completely means to write a polynomial as a product of factors that cannot be factored any further. First take out the greatest common factor, then factor what is left: 2x^2 − 18 = 2(x^2 − 9) = 2(x + 3)(x − 3). Stopping at 2(x^2 − 9) is not complete, because x^2 − 9 still factors.Algebra 1, Algebra 2
• FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x^2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Grade 6, Grade 7, Grade 8
• FOIL MethodThe FOIL method is a memory aid for multiplying two binomials: multiply the First terms, the Outer terms, the Inner terms and the Last terms, then combine like terms. For (x + 2)(x + 5): x^2 + 5x + 2x + 10 = x^2 + 7x + 10. It is the distributive property applied twice.Algebra 1
• Leading CoefficientThe leading coefficient of a polynomial is the number multiplying the term with the highest power of the variable. In 3x^2 − 5x^4 + 2x, the highest power is x^4, so the leading coefficient is −5. Write the polynomial in standard form first; its sign tells which way the graph's ends point.Algebra 1, Algebra 2
• MonomialA monomial is a single term made of a number, a variable, or a product of numbers and variables with whole-number exponents, such as 8, x, or −7x^2y. It has no addition or subtraction, and no variable in a denominator. Its degree is the sum of the exponents on its variables.Grade 8, Algebra 1
• PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x^2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra 1, Algebra 2
• Remainder TheoremThe Remainder Theorem says that when a polynomial p(x) is divided by x − a, the remainder equals p(a). So substituting x = a replaces the division. For p(x) = x^2 + 1 divided by x − 3, the remainder is p(3) = 10. A remainder of 0 means x − a is a factor.Algebra 2, Precalculus
• Synthetic DivisionSynthetic division is a shortcut for dividing a polynomial by a linear factor of the form x − k. You write only the coefficients, bring down the first one, then repeat two moves: multiply by k and add. The bottom row gives the quotient's coefficients, and its last number is the remainder, which equals the polynomial's value at k.Algebra 2, Precalculus
• TrinomialA trinomial is a polynomial with exactly three terms, such as x^2 + 5x + 6 or 2a − 3b + 7. The terms are joined by addition or subtraction and must be unlike terms, so nothing combines. The most common kind is the quadratic trinomial ax^2 + bx + c, which students learn to factor into two binomial factors.Algebra 1, Algebra 2

Linear equations 10

• EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Grade 6, Grade 7, Grade 8
• InequalityAn inequality compares values using symbols such as <, >, ≤ or ≥. Instead of saying two expressions are equal, it says one is greater, less, or at least/at most another, as in x > 5. An inequality with a variable usually has infinitely many solutions, shown as a ray on a number line.Grade 6, Grade 7, Grade 8
• Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Grade 7, Grade 8, Algebra 1
• Point-Slope FormPoint-slope form is a way to write the equation of a line from its slope m and one point (x1, y1) on it: y − y1 = m(x − x1). It is the fastest form to use when you know a point and the slope, or two points. Multiplying it out gives slope-intercept form.Grade 8, Algebra 1
• SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Grade 8, Algebra 1
• Slope-Intercept FormSlope-intercept form is the linear equation form y = mx + b, where m is the slope and b is the y intercept. You can read both values directly from the equation without rearranging: in y = 2x + 5, the slope is 2 and the line crosses the y-axis at (0, 5).Grade 8, Algebra 1
• SolutionA solution is a value that makes an equation or inequality true. To test a value, substitute it for the variable and check that the statement holds: x = 4 solves x + 2 = 6 because 4 + 2 = 6. A solution can also be an ordered pair, a point on a graph that satisfies the relationship.Grade 6, Grade 7, Grade 8
• Standard Form of a Linear EquationThe standard form of a linear equation is Ax + By = C, where A, B and C are integers and A and B are not both 0. Both variables sit on the left side. It makes the intercepts easy to find: the x-intercept is C/A, the y-intercept is C/B, and the slope is −A/B.Grade 8, Algebra 1
• X-InterceptAn x-intercept is a point where a graph crosses or touches the x axis. Its y-coordinate is always 0, so it is written (a, 0). To find it, set y = 0 and solve for x. A non-horizontal line has one x-intercept, while a parabola can have two, one or none.Grade 8, Algebra 1, Algebra 2
• Y-InterceptThe y-intercept is the point where a graph crosses the y-axis. Every point on the y-axis has x = 0, so you find the y-intercept by putting x = 0 into the equation. In y = mx + b, the y-intercept is b, written as the point (0, b).Grade 8, Algebra 1

Inequalities 1

• Compound InequalityA compound inequality is two inequalities joined by "and" or "or". With "and", a number must make both true, as in −2 < x ≤ 3, which is one segment on a number line. With "or", it only has to make at least one true, as in x < −1 or x ≥ 4, which is often two separate pieces.Algebra 1, Algebra 2

Systems of equations 3

• Elimination MethodThe elimination method solves a system of equations by adding or subtracting them so one variable cancels. Scaling one equation first is often needed to make the coefficients match. After one variable is eliminated, solve for the other variable, then substitute back to find the first.Grade 8, Algebra 1
• Substitution MethodThe substitution method solves a system of equations by replacing one variable with an equivalent expression. Solve one equation for a variable, put that expression into the other equation, and solve the new one-variable equation. Then substitute back to find the other variable. It is easiest when one equation is already solved for a variable.Grade 8, Algebra 1
• System of EquationsA system of equations is a set of two or more equations that use the same variables and are solved together. Its solution is the values that make every equation true at the same time. For two linear equations, that is the point where the two lines cross. A system can have one solution, none, or infinitely many.Grade 8, Algebra 1

Quadratics 9

• Completing the SquareCompleting the square is a method that rewrites x^2 + bx as a perfect square by adding (b/2)^2: x^2 + bx + (b/2)^2 = (x + b/2)^2. It is used to solve quadratic equations, to write a quadratic in vertex form, and to derive the quadratic formula.Algebra 1, Algebra 2, Precalculus
• DiscriminantThe discriminant of a quadratic equation ax^2 + bx + c = 0 is the number b^2 − 4ac. Its sign tells how many real solutions the equation has without solving it: positive means two, zero means one (a repeated root), and negative means none, because the parabola never touches the x-axis.Algebra 1, Algebra 2
• Quadratic EquationA quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra 1, Algebra 2
• Quadratic FormulaThe quadratic formula gives the solutions of any quadratic equation ax^2 + bx + c = 0 with a ≠ 0: x = (−b ± square root of (b^2 − 4ac))/2a. Substitute a, b and c, then compute once with plus and once with minus. It works even when the quadratic cannot be factored.Algebra 1, Algebra 2
• Real SolutionA real solution of an equation is a solution that is a real number, so it lies on the number line, such as x = 3 or x = −square root of 2. Some equations have none: x^2 + 4 = 0 has only imaginary solutions. For a quadratic, the discriminant b^2 − 4ac tells how many there are: 2, 1 or 0.Algebra 1, Algebra 2
• RootsThe roots of an equation are the values of the variable that make it true; for a polynomial equation set equal to 0, they are the solutions. The roots of x^2 + 3x − 10 = 0 are 2 and −5, because each makes the left side 0. Real roots appear as the x-intercepts of the graph.Algebra 1, Algebra 2
• Standard Form of a QuadraticThe standard form of a quadratic is ax^2 + bx + c, with the terms in descending powers of x and a ≠ 0. As an equation it is ax^2 + bx + c = 0; as a function, y = ax^2 + bx + c. It shows a, b and c directly, which the quadratic formula needs, and c is the y-intercept.Algebra 1, Algebra 2
• Vertex FormVertex form is a way to write a quadratic function as y = a(x − h)^2 + k, so the vertex (h, k) of its parabola can be read directly. The number a sets which way it opens and how wide it is. For example, y = 2(x − 3)^2 + 1 has its vertex at (3, 1).Algebra 1, Algebra 2
• Vertex of a ParabolaThe vertex of a parabola is its turning point: the lowest point when the parabola opens up, or the highest point when it opens down. For y = ax^2 + bx + c, the vertex has x-coordinate x = −b/2a; substitute that value to get the y-coordinate. The vertex sits on the axis of symmetry.Algebra 1, Algebra 2

Word problems 1

• Consecutive IntegersConsecutive integers are integers that follow each other in order, each 1 more than the one before, such as 7, 8, 9 or −2, −1, 0. In algebra, write them as n, n + 1, n + 2. Consecutive even or consecutive odd integers are 2 apart: n, n + 2, n + 4.Grade 7, Grade 8, Algebra 1

Matrices 1

• MatrixA matrix is a rectangular array of numbers arranged in rows and columns, written inside square brackets. A matrix with m rows and n columns has dimensions m × n, read "m by n". Each number is an entry, named by its row and then its column: a23 is in row 2, column 3. The plural is matrices.Algebra 2, Precalculus

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Updated November 2026·Reviewed by the Effortless Math team