CLEP College Algebra key terms

A letter or symbol, such as x, that stands for an unknown or changeable number in an expression or equation. Its value may be determined by solving or may vary freely.

A mathematical phrase combining numbers, variables, and operation signs, such as 3x + 5, that can be evaluated or simplified but contains no equals sign and so is not a statement of equality.

A statement asserting that two expressions have equal value, written with an equals sign. Solving it means finding every value of the variable that makes the statement true.

A sum of terms of the form ax^n, where each exponent n is a nonnegative whole number and each coefficient a is a real number. Polynomials have no variables in denominators or under radicals.

The largest exponent appearing on the variable in a single-variable polynomial. It caps the number of real roots and shapes the graph's long-run behavior.

The coefficient multiplying the highest-power term of a polynomial written in descending order. Its sign and size help predict the graph's end behavior and the count of possible positive roots.

Rewriting a polynomial or integer as a product of simpler pieces. For example, x^2 - 9 becomes (x - 3)(x + 3). Factoring is a primary tool for solving polynomial equations.

If a product of factors equals zero, then at least one factor must equal zero. This justifies setting each factor of a factored equation to zero and solving separately.

An equation equivalent to ax^2 + bx + c = 0 with a not equal to zero. It can be solved by factoring, completing the square, or the quadratic formula, and has at most two real solutions.

The rule x = (-b ± sqrt(b^2 - 4ac)) / (2a) giving all solutions of ax^2 + bx + c = 0. It works for every quadratic, even when factoring fails.

The quantity b^2 - 4ac from the quadratic formula. Its sign tells how many real solutions exist: positive gives two, zero gives one, and negative gives none in the real numbers.

A technique that rewrites a quadratic in the form a(x - h)^2 + k by adding and subtracting a constant. It reveals the vertex and also derives the quadratic formula.

A statement comparing two expressions with a symbol such as <, >, ≤, or ≥. Its solution is usually a range of values rather than a single number, often shown on a number line.

The distance of a number from zero on the number line, written |x|. It is never negative, and |x| = a with a > 0 gives two cases: x = a or x = -a.

A compact way to describe a set of real numbers using parentheses and brackets, such as (2, 5] for values greater than 2 and at most 5. Parentheses exclude endpoints; brackets include them.

The collection of all values that satisfy an equation or inequality. For an equation it may be one number, several numbers, or none; for an inequality it is typically an interval or union of intervals.

Equations that have exactly the same solution set. Adding the same quantity to both sides, or multiplying both sides by a nonzero constant, produces an equivalent equation.

A value produced during solving that does not actually satisfy the original equation, often arising after squaring both sides or clearing a variable denominator. Every candidate answer must be checked.

Two or more equations involving the same variables, solved together. A solution must satisfy every equation at once, which corresponds graphically to an intersection point of their graphs.

A way to solve a system by adding or subtracting multiples of the equations so one variable cancels. The goal is to reduce the system to a single equation in one variable.

A rule assigning exactly one output value to each input in its domain. Passing the vertical line test confirms that a graph represents a function.

The complete set of input values for which a function is defined. Common restrictions exclude inputs causing division by zero or a negative under an even root.

The set of all output values a function actually produces as its input runs through the domain. It is read from the vertical spread of the graph.

A graphical check: a curve represents a function only if no vertical line meets it more than once. A second intersection would mean one input has two outputs.

A graphical check for whether a function is one-to-one: if no horizontal line crosses the graph more than once, the function has an inverse that is also a function.

A function in which different inputs always give different outputs, so no two domain values share an output. Only such functions possess a true inverse function.

The function that reverses another, undoing its operation, written f^-1. It satisfies f(f^-1(x)) = x, and its graph is the original reflected across the line y = x.

A function formed by applying one function to the output of another, written (f ∘ g)(x) = f(g(x)). The inner function's range must fit inside the outer function's domain.

A function of the form f(x) = mx + b, whose graph is a straight line. The slope m measures the constant rate of change and b is the y-intercept.

The steepness of a line, computed as the change in y divided by the change in x between two points. It is the constant rate of change of a linear function.

How much one quantity changes per unit change in another, calculated as a difference in outputs divided by the corresponding difference in inputs. For a line it is constant and equals the slope.

A function of the form f(x) = ax^2 + bx + c with a not equal to zero. Its graph is a parabola opening upward when a is positive and downward when a is negative.

The highest or lowest point of a parabola, where it changes direction. For f(x) = ax^2 + bx + c it occurs at x = -b/(2a).

The vertical line through a parabola's vertex that divides the graph into mirror images. Its equation is x = -b/(2a) for a quadratic in standard form.

What happens to a function's output values as the input grows very large positive or very large negative. For polynomials it is governed by the degree and leading coefficient.

A fraction whose numerator and denominator are polynomials, such as (x + 1)/(x - 3). It is undefined wherever the denominator equals zero, which creates domain restrictions.

An equation containing a variable inside a square root or other root, such as sqrt(x + 4) = x. Solving usually requires isolating the radical and squaring both sides.

A function of the form f(x) = a · b^x with base b positive and not equal to one. It models constant percentage growth or decay rather than constant additive change.

The exponent to which a base must be raised to produce a given number, written log_b(x). It is the inverse relationship of the exponential function b^x and turns products into sums.

A number of the form a + bi, where a and b are real and i is the imaginary unit with i^2 = -1. Complex numbers let every quadratic equation have two solutions.

Any number on the continuous number line, including rationals such as 3/4 and irrationals such as sqrt(2). Real numbers can be ordered and include every decimal expansion.

A real number that cannot be written as a fraction of two integers. Its decimal expansion never terminates or repeats, as with pi and sqrt(2).

A number expressible as a ratio of two integers with a nonzero denominator. Its decimal form either terminates or eventually repeats a block of digits.

The agreed sequence for evaluating expressions: parentheses first, then exponents, then multiplication and division left to right, then addition and subtraction left to right. It guarantees a single correct value.

Rules governing powers, such as x^m · x^n = x^(m+n), (x^m)^n = x^(mn), and x^-n = 1/x^n. They allow expressions with powers to be simplified and rewritten.