Algebra is the language of unknowns. A variable, usually a letter like \(x\) or \(\theta\), stands in for a number we may or may not know, and an expression combines variables with the standard operations. Evaluating an expression means substituting known values for variables and simplifying using the order of operations (parentheses, exponents, multiplication and division from left to right, then addition and subtraction). Like terms share the same variable part, so \(3x^2\) and \(-7x^2\) combine freely while \(3x^2\) and \(3x\) do not; combining means adding or subtracting coefficients, never exponents.
The distributive property, \(a(b+c) = ab + ac\), is the engine that powers both expansion and factoring. To expand, you distribute a factor across every term inside parentheses; to factor, you reverse the process, pulling out the greatest common factor. The FOIL pattern (First, Outer, Inner, Last) is a specialized distributive technique for multiplying two binomials. Several factoring identities appear constantly: \(a^2 - b^2 = (a+b)(a-b)\) (difference of squares), \((a \pm b)^2 = a^2 \pm 2ab + b^2\) (perfect square trinomials), and \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\) (sum or difference of cubes). For general trinomials \(ax^2 + bx + c\), you find two numbers whose product is \(ac\) and whose sum is \(b\), then split the middle term and factor by grouping.
Solving an equation means isolating the variable using inverse operations while keeping both sides balanced. Linear equations collapse to a single value through simple arithmetic; systems of two linear equations can be solved by substitution or by adding the equations after multiplying so that one variable cancels. Quadratic equations can be solved by factoring (and applying the zero product property, which says that if a product equals zero then at least one factor must be zero), by completing the square (adding \((b/2)^2\) to form a perfect square), or by the quadratic formula \(x = (-b \pm \sqrt{b^2-4ac})/(2a)\). The discriminant \(b^2 - 4ac\) tells you at a glance whether there are two real roots, one repeated root, or two complex roots.
Rational expressions, which are ratios of polynomials, behave much like numerical fractions. You multiply them by multiplying numerators and denominators, divide by flipping the second fraction, and add or subtract them by finding a common denominator. Simplifying means factoring and canceling, while remembering that values making the denominator zero are excluded from the domain. Beyond solving, algebra also studies polynomial structure: a polynomial's degree is its highest power, the Fundamental Theorem of Algebra guarantees \(n\) roots for a degree-\(n\) polynomial (counting multiplicity), and polynomials can be added, subtracted, multiplied, and divided using long division. These tools build up from arithmetic and form the basis for nearly every higher idea in mathematics.