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Algebra
Equations, functions, polynomials, and the language used to describe mathematical structure.
Start with Absolute Value Equations →74 complete · 74 total lessons
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01Absolute Value EquationsAn absolute value equation describes a distance condition. If $|u|=k$ with $k>0$, then $u=k$ or $u=-k$ because two points lie distance $k$ from zero.Full lesson + sheet02Absolute Value InequalitiesAbsolute value inequalities describe distance ranges. For $k>0$, $|u|<k$ means $-k<u<k$ (inside a band), while $|u|>k$ means $u<-k$ or $u>k$ (outside the band).Full lesson + sheet03Adding PolynomialsAdding polynomials combines like terms: terms with identical variables raised to identical exponents. The operation adds coefficients while preserving each common variable part.Full lesson + sheet04Adding Rational ExpressionsA rational expression is a quotient of polynomials. Addition requires a common denominator because numerators count units of that denominator.Full lesson + sheet05Arithmetic SequencesAn arithmetic sequence changes by a constant difference and can be described recursively or with a linear explicit formula.Full lesson + sheet06Average Rate of ChangeAverage rate of change measures output change per input change across an interval and equals the slope of a secant line.Full lesson + sheet07Axis of SymmetryAn axis of symmetry is a line that divides a figure or graph into mirror-image halves. For a vertical parabola $y=ax^2+bx+c$ with $a\ne0$, its axis is $x=-\frac{b}{2a}$.Full lesson + sheet08BinomialA binomial is a polynomial with exactly two nonzero unlike terms after simplification. Examples include $x+5$, $3a^2-2a$, and $p^3q+7$.Full lesson + sheet09Binomial TheoremThe binomial theorem expands a nonnegative integer power of a sum: $(a+b)^n=\sum_{k=0}^n\binom nk a^{n-k}b^k$.Full lesson + sheet10Completing the SquareCompleting the square rewrites a quadratic as a perfect square plus a constant, revealing its vertex and solutions.Full lesson + sheet11Complex FractionsA complex fraction has a fraction in its numerator, denominator, or both. It is division of one rational expression by another.Full lesson + sheet12Complex NumbersA complex number has form $a+bi$, where $a,b$ are real and $i^2=-1$. The real part is $a$ and imaginary part is $b$.Full lesson + sheet13Degree of a PolynomialFor a nonzero polynomial in one variable, the degree is the greatest exponent with a nonzero coefficient after simplification.Full lesson + sheet14Difference of SquaresA difference of squares is a binomial $a^2-b^2$ that factors as $(a-b)(a+b)$. The identity follows because the middle terms cancel during expansion.Full lesson + sheet15DiscriminantThe discriminant predicts the number and type of quadratic roots before the equation is fully solved.Full lesson + sheet16Dividing Rational ExpressionsDividing rational expressions uses the reciprocal of the divisor: $A/B\div C/D=A/B\cdot D/C$.Full lesson + sheet17Exponential EquationsExponential equations place the unknown in an exponent and can be solved with common bases, graphs, or logarithms.Full lesson + sheet18Exponential FunctionsExponential functions model repeated multiplication, including percent growth, decay, doubling, and half-life.Full lesson + sheet19Factor TheoremThe factor theorem connects a polynomial zero P(k)=0 with the linear factor x−k.Full lesson + sheet20Factored FormFactored form y=a(x−r)(x−s) reveals a quadratic's zeros, symmetry, opening, and scale.Full lesson + sheet21FactoringFactoring rewrites an expression as a product, revealing common structure, zeros, simplifications, and efficient algebraic methods.Full lesson + sheet22Factoring by GroupingFactoring by grouping rewrites a polynomial as groups with a shared binomial or other common factor. For four terms, pair terms, factor the GCF from each pair, then factor the repeated bracket.Full lesson + sheet23Factoring TrinomialsFactoring trinomials finds binomial factors whose outer and inner products rebuild the middle term.Full lesson + sheet24Function CompositionFunction composition sends an input through one function and then uses that output as the input of another.Full lesson + sheet25Function TransformationsFunction transformations move, stretch, compress, and reflect a familiar graph without rebuilding it point by point.Full lesson + sheet26Geometric SequencesA geometric sequence changes by a constant ratio and forms a discrete exponential pattern.Full lesson + sheet27Greatest Common Factor FactoringGreatest common factor factoring extracts the largest monomial dividing every term of a polynomial. It reverses distribution: $ab+ac=a(b+c)$.Full lesson + sheet28Instantaneous Rate of ChangeInstantaneous rate of change measures a function's rate at one input and equals the slope of its tangent line.Full lesson + sheet29Inverse FunctionsAn inverse function reverses a one-to-one function by exchanging inputs and outputs.Full lesson + sheet30Linear EquationSolve a linear equation by preserving equality while undoing the operations around the variable.Full lesson + sheet31Linear Inequalities in Two VariablesA linear inequality in two variables describes a half-plane of solutions separated by a boundary line. Points on the boundary are included for $\le$ or $\ge$ and excluded for $<$ or $>$.Full lesson + sheet32Linear RelationsA linear relation has a constant rate of change and can be represented by a table, graph, equation, or context.Full lesson + sheet33Logarithmic EquationsLogarithmic equations are solved by respecting positive arguments, combining logs legally, and using their inverse exponential relationship.Full lesson + sheet34Logarithmic FunctionsA logarithm answers an exponent question and forms the inverse of an exponential function.Full lesson + sheet35MonomialA monomial is a single polynomial term: a constant times variables raised to nonnegative integer exponents.Full lesson + sheet36Multiplying PolynomialsMultiplying polynomials applies the distributive property so every term of one factor multiplies every term of the other.Full lesson + sheet37Multiplying Rational ExpressionsMultiplying rational expressions multiplies numerators and denominators, but factoring first exposes common factors that can be cancelled.Full lesson + sheet38ParabolaA parabola is a symmetric curve produced by a quadratic function and defined geometrically by equal distance from a focus and directrix.Full lesson + sheet39Parallel LinesParallel lines keep a constant separation and, in the coordinate plane, have equal direction and equal slopes when nonvertical.Full lesson + sheet40Perfect Square TrinomialsA perfect square trinomial is produced by squaring a binomial: $a^2+2ab+b^2=(a+b)^2$ or $a^2-2ab+b^2=(a-b)^2$.Full lesson + sheet41Perpendicular LinesPerpendicular lines meet at a right angle; their nonzero finite slopes are negative reciprocals.Full lesson + sheet42Piecewise FunctionsA piecewise function uses different formulas on different parts of its domain. The input condition selects exactly which rule to apply.Full lesson + sheet43Point-Slope FormPoint-slope form $y-y_1=m(x-x_1)$ describes the line with slope $m$ through $(x_1,y_1)$. It comes from the slope relation $(y-y_1)/(x-x_1)=m$ and is especially useful when one point and slope are known.Full lesson + sheet44PolynomialA polynomial combines constant multiples of whole-number powers into one of algebra's most useful families.Full lesson + sheet45Polynomial FunctionsPolynomial functions combine powers of x with nonnegative integer exponents and have smooth graphs shaped by degree, zeros, and leading coefficient.Full lesson + sheet46Polynomial InequalitiesPolynomial inequalities are solved by locating zeros and determining where the polynomial is positive or negative.Full lesson + sheet47Quadratic EquationA quadratic equation contains a squared variable and asks for the values that make its related parabola reach zero.Full lesson + sheet48Quadratic FormulaThe quadratic formula solves every quadratic equation and reveals how its roots depend on the shape of its parabola.Full lesson + sheet49Quadratic FunctionA quadratic function has constant nonzero second differences and graphs as a parabola whose forms reveal zeros, vertex, and intercepts.Full lesson + sheet50Radical ExpressionsA radical expression contains a root such as $\sqrt[n]{a}$. Simplification extracts perfect $n$th-power factors while preserving exact value.Full lesson + sheet51Rational EquationsRational equations contain variable denominators and are solved by recording restrictions, clearing denominators, and checking candidates.Full lesson + sheet52Rational ExponentsFor a positive real base $a$ and integers $m,n$ with $n>0$, a rational exponent represents roots and powers: $a^{m/n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$.Full lesson + sheet53Rational ExpressionsRational expressions are polynomial fractions whose algebra is governed by factoring, common denominators, and domain restrictions.Full lesson + sheet54Rational FunctionsRational functions divide polynomials and may have holes, asymptotes, intercepts, and separate branches.Full lesson + sheet55Rational InequalitiesRational inequalities use zeros, undefined values, and sign charts to determine where a quotient is positive or negative.Full lesson + sheet56Rational Root TheoremThe rational root theorem lists possible rational zeros of an integer-coefficient polynomial.Full lesson + sheet57Remainder TheoremWhen P(x) is divided by x−k, the remainder is the single value P(k).Full lesson + sheet58Roots and ZerosA zero of a function is an input $r$ for which $f(r)=0$. For a polynomial, zero, root of $f(x)=0$, $x$-intercept (when real), and factor $x-r$ describe linked aspects of the same value.Full lesson + sheet59SeriesA series adds the terms of a sequence; arithmetic and geometric structures provide efficient finite-sum formulas.Full lesson + sheet60Simplifying Rational ExpressionsSimplifying a rational expression factors numerator and denominator and cancels common nonzero factors. It does not cancel terms joined by addition.Full lesson + sheet61SlopeSlope measures a line’s vertical change per unit of horizontal change and represents a constant rate of change.Full lesson + sheet62Slope-Intercept FormSlope-intercept form writes a linear relationship as y = mx + b, exposing its constant rate of change and starting value.Full lesson + sheet63Solving Systems by EliminationElimination combines equivalent equations so one variable cancels, leaving a one-variable equation.Full lesson + sheet64Solving Systems by GraphingSolving a linear system by graphing identifies the shared point or shared set of its lines.Full lesson + sheet65Solving Systems by SubstitutionSubstitution solves a system by replacing one variable with an equivalent expression from the other equation.Full lesson + sheet66Special ProductsSpecial products are recurring polynomial multiplication patterns: $(a+b)^2=a^2+2ab+b^2$, $(a-b)^2=a^2-2ab+b^2$, and $(a+b)(a-b)=a^2-b^2$.Full lesson + sheet67Standard Form of a LineStandard form Ax + By = C displays a linear equation with both variables aligned and makes intercepts, elimination, and integer structure convenient.Full lesson + sheet68Subtracting PolynomialsSubtracting a polynomial means adding its additive inverse. The negative sign before a bracket changes every term inside; after distribution, like terms combine by adding signed coefficients.Full lesson + sheet69Synthetic DivisionSynthetic division is a compact algorithm for dividing a polynomial by a linear divisor $x-c$. It operates on coefficients and produces quotient coefficients plus remainder $P(c)$.Full lesson + sheet70Systems of Linear EquationsA system of linear equations asks for values that satisfy multiple linear relationships at the same time.Full lesson + sheet71TrinomialA trinomial is a polynomial with exactly three nonzero unlike terms after simplification. Term count does not specify degree: $x^2+3x+2$ is quadratic, while $a^7-a+1$ is a degree-seven trinomial.Full lesson + sheet72Variation and ModelingVariation models express how quantities change together through direct, inverse, joint, and power relationships.Full lesson + sheet73VertexThe vertex of a parabola is its turning point and lies on the axis of symmetry. For $y=a(x-h)^2+k$, the vertex is $(h,k)$.Full lesson + sheet74Vertex FormVertex form makes a parabola's turning point, axis of symmetry, opening, and vertical scale immediately visible.Full lesson + sheet
