College Algebra Vocabulary Review

Vocabulary flashcards covering core algebra concepts including polynomial equations, complex numbers, conics, functions, number systems, and coordinate formulas based on the study guide.

Complex Conjugates Theorem

A theorem that informally guarantees that complex roots of a polynomial or quadratic equation always come in pairs: $(a+bi)$ and $(a-bi)$.

Radical Equations

Equations that contain variables in the expression under a radical symbol.

Extraneous Solutions

Solutions obtained during the process of solving an equation (such as a radical equation) that do not satisfy the original equation and must be rejected.

Fundamental Theorem of Algebra

States that a polynomial equation of degree $n$ has exactly $n$ roots, which may be real or complex.

Roots of a Quadratic Equation

The algebraic solutions obtained when solving a quadratic equation.

Zeros of a Function

The $x$-intercepts of a function, representing the points where the graph intersects the $x$-axis.

Discriminant

The expression $b^2 - 4ac$ used to determine the number and nature of solutions for a quadratic equation: 2 real solutions if greater than 0, 1 real solution if equal to 0, and 2 complex solutions if less than 0.

Rational Exponents

Exponents written as fractions, where the numerator represents a power and the denominator represents a root.

Rational Expression

The ratio, or quotient, of two polynomials.

Rational Equation

An equation containing at least one rational expression with a variable in the denominator of at least one term.

Complex Numbers

The set of numbers that includes all Real Numbers plus all Imaginary Numbers, written in the format $a+bi$.

Imaginary Numbers

Numbers whose square is negative, defined using $i = \sqrt{-1}$ and $i^2 = -1$.

Real Numbers

The set of numbers consisting of all Rational Numbers plus all Irrational Numbers.

Rational Numbers

The set of numbers that can be written as a simplified fraction, including repeating decimals and terminating decimals.

Irrational Numbers

The set of numbers that cannot be written as a simplified fraction, including $\pi$ and radicals that do not contain perfect squares such as $\sqrt{8}$.

Integers

The set of numbers $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

Whole Numbers

The set of non-negative integers \object{\{0, 1, 2, 3, \dots\}}.

Natural Numbers

Also known as counting numbers, the set of positive integers \object{\{1, 2, 3, \dots\}}.

Empty Set

Represented as \object{\{\}}, a set containing no elements.

Ellipse Parameter $a$

In the standard equation of an ellipse, the value that indicates the amount of horizontal stretch or shrink.

Ellipse Parameter $b$

In the standard equation of an ellipse, the value that indicates the amount of vertical stretch or shrink.

Ellipse Parameter $k$

In the formula for an ellipse, the value that indicates the amount of vertical translation (up or down).

Even Function

A function whose graph has the $y$-axis as its axis of symmetry, verified algebraically using the formula $f(-x) = f(x)$.

Odd Function

A function whose graph is symmetric about the origin, verified algebraically using the formula $f(-x) = -f(x)$.

Cartesian Coordinate System

The rectangular coordinate system created by René Descartes in 1637.

Distance Formula

The formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ used to find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$.

Midpoint Formula

The formula $\text{midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ used to calculate the center point between two coordinates.