DSST Fundamentals of College Algebra key terms

The number whose square equals negative one, written i, so that i² = −1. It extends the real numbers and lets equations such as x² + 1 = 0 have solutions.

A number written a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a and the imaginary part is b.

For z = a + bi, the real part is a and the imaginary part is the real number b (not bi). Two complex numbers are equal exactly when both parts match.

For z = a + bi, the conjugate is a − bi, written with a bar over z. Multiplying a complex number by its conjugate gives the real number a² + b².

The distance from the origin to the point representing a + bi in the complex plane, equal to the square root of a² + b². It is never negative.

A coordinate plane where the horizontal axis shows real parts and the vertical axis shows imaginary parts. Each complex number a + bi corresponds to exactly one point (a, b).

For a positive real number k, the principal square root of −k is written i times the square root of k. This convention keeps square-root expressions consistent when imaginary numbers appear.

Rewriting a quotient of complex numbers so no i remains in the denominator, done by multiplying numerator and denominator by the conjugate of the denominator.

The values of i raised to whole-number exponents repeat in a cycle of four: i, −1, −i, 1. Dividing the exponent by 4 and using the remainder identifies the value.

A variable is a symbol, usually a letter, standing for a value that may change; a constant is a fixed value. Expressions combine them with operations.

A combination of variables, constants, and operations such as addition, subtraction, multiplication, division, and exponentiation, with no equals sign. Examples include 3x² − 5x + 1.

The numerical factor multiplying a variable or power of a variable in a term. In −7x³ the coefficient is −7; in x it is understood to be 1.

A term is a single product of numbers and variables, such as 4x²y. Like terms have identical variable parts and can be combined by adding their coefficients.

The largest exponent on the variable in a single-variable polynomial, or the largest sum of exponents in any one term of a multivariable polynomial. It helps predict the number of roots.

A sum of terms of the form axⁿ where n is a whole number and a is a real coefficient. Polynomials are defined for all real numbers and have no variables in denominators or under radicals.

Rewriting a polynomial as a product of simpler polynomials. Common patterns include greatest common factor, difference of squares, and trinomial factoring.

The pattern a² − b² = (a + b)(a − b). It factors any binomial whose two terms are perfect squares separated by subtraction.

A trinomial matching a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a − b)². Recognizing the pattern speeds up factoring and completing the square.

A fraction whose numerator and denominator are polynomials. Its domain excludes values making the denominator zero, and it is simplified by canceling common factors.

Rewriting a fraction so no radical remains in the denominator, usually by multiplying by a form of one built from the radical or its conjugate.

An equation in which every variable appears to the first power, so its graph is a straight line. A single-variable linear equation has exactly one solution unless it is an identity or a contradiction.

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

An equation of the form ax² + bx + c = 0 with a not zero. It can be solved by factoring, completing the square, or the quadratic formula, and has at most two real solutions.

The solution formula x = (−b ± √(b² − 4ac)) / (2a) for ax² + bx + c = 0 with a ≠ 0. It solves every quadratic equation, including those with no real roots.

The quantity b² − 4ac from the quadratic formula. If positive there are two distinct real roots, if zero one repeated real root, and if negative two nonreal complex roots.

Rewriting ax² + bx + c by adding and subtracting (b/2)² to create a perfect square trinomial. It produces the vertex form of a quadratic and can solve quadratic equations.

If a product of factors equals zero, then at least one factor equals zero. It converts a factored equation such as (x − 3)(x + 5) = 0 into simpler equations to solve.

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

An equation in which the variable appears under a radical. Isolating the radical and raising both sides to a power solves it, but the step can introduce extraneous solutions.

An equation containing an absolute value expression. An equation of the form |expression| = k with k positive splits into two cases; with k negative it has no solution.

An inequality in which variables appear only to the first power. Multiplying or dividing both sides by a negative number reverses the direction of the inequality symbol.

Two inequalities joined by and or or. An and statement requires both conditions and gives an intersection; an or statement requires either and gives a union.

An inequality involving absolute value. |x| < k means the distance from x to zero is under k, giving a between statement; |x| > k gives a beyond statement with two pieces.

A compact way to write a set of real numbers using parentheses for excluded endpoints and brackets for included ones. For example, [−2, 5) means −2 ≤ x < 5.

A rule assigning each input in its domain exactly one output. The vertical line test determines whether a graph represents a function.

The set of all inputs for which a function is defined. It excludes values causing division by zero or an even root of a negative number.

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

A graph represents a function exactly when no vertical line crosses it more than once. A second crossing would mean one input has two outputs.

A function is one-to-one exactly when no horizontal line crosses its graph more than once. Passing the test means the function has an inverse that is also a function.

The function that reverses another, written f⁻¹, so that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Its graph is the reflection of the original graph across the line y = x.

The steepness of a line, computed as the change in y divided by the change in x between two points. Parallel lines share a slope; perpendicular slopes multiply to −1.

The form f(x) = a(x − h)² + k, where (h, k) is the vertex of the parabola. The sign of a tells whether it opens upward or downward.

The vertical line x = h through the vertex of a parabola, dividing it into mirror-image halves. For f(x) = ax² + bx + c it is x = −b/(2a).

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.

A function formed by applying one function to the output of another, written (f ∘ g)(x) = f(g(x)). The domain requires every intermediate output to be valid.

Every polynomial equation of degree n with n at least 1 has at least one complex root, and counting multiplicity it has exactly n complex roots.