CLEP College Algebra flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Difference of squares

The factoring pattern a² − b² = (a − b)(a + b). It applies whenever two perfect squares are subtracted, and it is the fastest route to factoring expressions like x² − 49 or 9y² − 16.

Zero exponent rule

For any nonzero base a, a⁰ = 1. The rule follows from dividing a^n by itself, since a^n/a^n = a^(n−n) = a⁰. The expression 0⁰ is undefined.

Negative exponent rule

a^(−n) = 1/a^n for a ≠ 0, so a negative exponent signals a reciprocal, not a negative value. For example, 2^(−3) = 1/8 and (3/4)^(−2) = 16/9.

Rational exponent equivalence

a^(m/n) equals the n-th principal root of a^m, which also equals the n-th root of a raised to the m-th power. The radicand must be nonnegative when n is even.

Sum of cubes

a³ + b³ factors as (a + b)(a² − ab + b²). The middle term is subtracted and the sign pattern is SOAP: same, opposite, always positive. The similar-looking difference a³ − b³ uses (a − b)(a² + ab + b²).

Conjugate rationalization

To remove a radical from a denominator like 1/(a + √b), multiply numerator and denominator by a − √b. The denominator becomes a² − b, a rational number.

Discriminant test

For ax² + bx + c = 0, the quantity b² − 4ac determines the roots: positive means two distinct real roots, zero means one repeated real root, negative means no real roots.

Quadratic formula

The solutions of ax² + bx + c = 0 with a ≠ 0 are x = [−b ± √(b² − 4ac)]/(2a). It solves every quadratic, including ones that do not factor over the integers.

Absolute value inequality

For k > 0, |u| < k is equivalent to −k < u < k, a conjunction. By contrast |u| > k splits into u < −k or u > k, a disjunction. Equality |u| = k gives u = ±k.

Extraneous solution

A value produced while solving a rational or radical equation that fails when substituted into the original equation, usually because it makes a denominator zero or a radicand negative. Such values must be discarded.

Inequality reversal rule

Multiplying or dividing both sides of an inequality by a negative number flips the inequality symbol. Adding or subtracting the same quantity on both sides never changes the direction.

Zero-product property

If a product of factors equals zero, at least one factor equals zero. Solving by factoring therefore requires moving all terms to one side so the other side is zero before setting factors equal to zero.

System with no solution

Two linear equations whose graphs are parallel distinct lines have no common point, so the system is inconsistent. Algebraically, elimination produces a false statement such as 0 = 5.

Interval notation

A bracket, as in [2, 7], includes the endpoint; a parenthesis, as in (2, 7), excludes it. Infinity always takes a parenthesis, and a union symbol joins disjoint solution pieces.

Vertical line test

A graph represents a function exactly when no vertical line crosses it more than once. A single vertical line hitting two points means one input has two outputs, which violates the function definition.

Horizontal line test

A function is one-to-one, and therefore has an inverse function, exactly when every horizontal line meets its graph at most once. Failing the test means the inverse would not be a function.

Composition of functions

(f ∘ g)(x) = f(g(x)) means apply g first, then f to that result. Composition is generally not commutative, and the domain of f ∘ g excludes inputs where g(x) is outside f's domain.

Inverse function procedure

To find f⁻¹, replace f(x) with y, swap x and y, then solve for y. The result satisfies f(f⁻¹(x)) = x, and its graph is the reflection of f across the line y = x.

Vertex of a parabola

For y = ax² + bx + c, the vertex's x-coordinate is −b/(2a). The vertex is the minimum when a > 0 and the maximum when a < 0, and the axis of symmetry is that vertical line.

Domain of a rational function

The domain is all real numbers except those making the denominator zero. After factoring, set each denominator factor equal to zero and exclude those solutions; numerator zeros do not restrict the domain.

Even versus odd function

An even function satisfies f(−x) = f(x) and is symmetric about the y-axis. An odd function satisfies f(−x) = −f(x) and is symmetric about the origin. Many functions are neither.

Average rate of change

The average rate of change of f on the interval from a to b is [f(b) − f(a)]/(b − a), the slope of the secant line through those two graph points.

Exponential growth model

y = a·b^x describes growth when b > 1 and decay when 0 < b < 1, with a the initial value. Compound interest A = P(1 + r/n)^(nt) is a standard application.

Vertical shift transformation

Adding a constant outside the function, y = f(x) + k, moves the graph up by k units when k is positive and down when k is negative. The shape and orientation are unchanged.

Rational number definition

A rational number can be written as p/q where p and q are integers and q ≠ 0. Its decimal form terminates or repeats. Integers and terminating decimals are therefore rational.

Irrational number definition

An irrational number cannot be written as a ratio of integers, so its decimal expansion is nonterminating and nonrepeating. Familiar examples include √2, √3, π, and e.

Closure property

A set is closed under an operation when performing that operation on members always yields a member of the same set. Integers are closed under addition and multiplication but not division.

Order of operations

Evaluate grouping symbols first, then exponents and radicals, then multiplication and division from left to right, and finally addition and subtraction from left to right. Multiplication does not automatically precede division.

Distributive property

a(b + c) = ab + ac states that multiplication distributes over addition or subtraction. It justifies expanding expressions and is the reverse of factoring out a common factor.

Absolute value as distance

|a| is the distance between a and zero on the number line, so it is never negative. Consequently |−a| = |a|, and |a − b| measures the distance between a and b.

Scientific notation rules

A number is written as a × 10^n with 1 ≤ |a| < 10. Multiplying adds exponents, dividing subtracts them, and raising a power to a power multiplies the exponent by that power.

Imaginary unit i

The imaginary unit satisfies i² = −1, so i = √(−1). Powers of i cycle through i, −1, −i, 1 every four exponents, and complex conjugates a ± bi multiply to the real value a² + b².