Basic Algebra

Elementary algebra from the ground up — variables, equations, lines, polynomials, factoring, fractions, and radicals, building to the Fundamental Theorem of Algebra.

Foundations

• Variables & Expressions — Letters stand for numbers — translate, evaluate, and combine like terms.
• Order of Operations — PEMDAS: the agreed order that makes every expression unambiguous.
• Signed Numbers — Add, subtract, multiply, and divide positives and negatives.
• Types of Numbers — Integers, rationals, irrationals, and the reals — and what 'real roots' means.
• Fractions, Decimals & Percents — One number, three notations — reduce and convert freely among them.

Equations, Functions & Applications

• Linear Equations — Use inverse operations to find the values that make an equation true.
• Literal Equations — Solve a formula for any one of its letters.
• Ratios & Proportions — Equal ratios and the cross-multiplication that solves them.
• Functions & Function Notation — Evaluating f(x), the single-output rule, domain, and range.
• Direct & Inverse Variation — y=kx grows together; xy=k trades off.
• Percent & Mixture Problems — Percent of a number, percent change, simple interest, and mixtures.
• The Coordinate Plane — (x, y) — over, then up. Every point gets an address.
• Slope of a Line — Rise over run — the steepness of a line between two points.
• Graphing Lines — Slope-intercept form and reading a line from its equation.
• Systems of Equations — Two equations, two unknowns — by substitution and elimination.
• Word Problems — Turn a sentence into an equation, then solve it.

Inequalities

• Linear Inequalities in One Variable — Solve and graph one-variable inequalities — and flip the sign on a negative.
• Compound Inequalities — Join inequalities with and / or, then write the interval.
• Absolute Value Equations & Inequalities — Distance from zero: split equations into two cases and inequalities into a band or two rays.
• Linear Inequalities in Two Variables — Boundary line, solid or dashed, then shade the right half-plane.
• Systems of Linear Inequalities — Intersect half-planes to find the feasible region.

Exponents, Polynomials & Factoring

• Exponent Rules — Product adds exponents, quotient subtracts, power of a power multiplies.
• Negative & Zero Exponents — Why a^0 = 1 and a^-n = 1/a^n, and rewriting with positive exponents.
• Scientific Notation — Write numbers as a × 10^n and compute with them.
• Polynomial Basics — Degree, leading coefficient, constant term, roots — the vocabulary of polynomials.
• Polynomial Expansion (FOIL) — (x + a)(x + b) = x^2 + (a + b)x + ab — the FOIL pattern.
• Polynomial Division — Divide by a monomial, then long-divide one polynomial by another.
• Special Products — Binomial squares and the cross term you must not drop.
• Difference of Squares — Spot the pattern A^2 - B^2 and split it in two.
• Factoring Polynomials — Reverse distribution: pull out common factors, then group.
• Factoring Trinomials — Reverse FOIL: split a trinomial back into two binomials.

Fractions, Radicals & Distance

• Rational Expressions — Algebraic fractions: reduce, multiply, divide, add, and subtract.
• Radicals & Roots — Simplify square roots and rationalize a denominator.
• Midpoint & Distance — Find the midpoint and the distance between two points.
• Rational Exponents — a^m/n means root-then-power: (√[n]a)^m.
• Radical Equations — Isolate the radical, square both sides, and check for extraneous roots.

Quadratics & Beyond

• Solving Quadratics by Factoring — Set it to zero, factor, and use the zero-product property.
• Completing the Square — x^2 + bx = (x + b/2)^2 - (b/2)^2 — make a perfect square.
• Quadratic Equation — The quadratic formula x = -b ± √b^2 - 4ac2a solves ax^2 + bx + c = 0.
• Complex Numbers — i^2 = -1 extends ℝ to ℂ. Conjugates make division work.
• Fundamental Theorem of Algebra — Every non-constant polynomial has a complex root — ℂ is algebraically closed.

Not sure where to start? Take the ten-question placement test.