FAMAT Algebra 2 Study Guide Flashcards

A comprehensive set of vocabulary flashcards covering the FAMAT Algebra 2 Study Guide topics including real numbers, equations, functions, polynomials, conics, sequences, combinatorics, and matrices.

Natural Numbers

The set of numbers denoted by $N = \{1, 2, 3, \dots\}$.

Rational Numbers ($Q$)

Numbers that can be expressed in the form $\frac{p}{q}$ where $p, q \in Z$ and $q \neq 0$.

Literal Equations

Equations where you solve for a specific variable, such as solving $V = \pi r^2 h$ for $h = \frac{V}{\pi r^2}$.

Conjunction (AND)

A compound inequality of the form $a < x < b$.

Slope-Intercept Form

The linear equation form $y = mx + b$.

Point-Slope Form

The linear equation form $y - y_1 = m(x - x_1)$.

Vertical Line Test

A test used to determine if a graph is a function; if a vertical line crosses the graph more than once, it is not a function.

Composite Functions

A function operation denoted as $(f \circ g)(x) = f(g(x))$ where the inner function $g(x)$ is evaluated first.

Holes (Removable Discontinuities)

Points in a rational function where the numerator $P(x)$ and denominator $Q(x)$ share a common factor that cancels.

Even Functions

Functions that satisfy $f(-x) = f(x)$ and are symmetric about the $y$-axis.

Odd Functions

Functions that satisfy $f(-x) = -f(x)$ and are symmetric about the origin.

Determinant ($2 \times 2$)

For a matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, it is calculated as $ad - bc$.

Imaginary Unit ($i$)

A unit defined as $i = \sqrt{-1}$, where $i^2 = -1$.

Modulus (Magnitude) of $z = a + bi$

The value calculated as $|z| = \sqrt{a^2 + b^2}$.

Monic Polynomial

A polynomial in which the leading coefficient (the coefficient of the highest-degree term) is $1$.

Multiplicity

Refers to how many times a root appears as a factor in a polynomial; for $(x - r)^m$, the root $r$ has a multiplicity of $m$.

Remainder Theorem

The theorem stating that the remainder of a polynomial $f(x)$ divided by $x - c$ is exactly $f(c)$.

Fundamental Theorem of Algebra

The theorem stating that every polynomial of degree $n \geq 1$ has $n$ roots in the complex numbers $C$ when counting multiplicity.

Rational Root Theorem

A theorem stating that if a polynomial has rational roots $\frac{p}{q}$, then $p$ must divide the constant term and $q$ must divide the leading coefficient.

Descartes' Rule of Signs

A rule stating that the number of sign changes in $f(x)$ is the maximum number of positive real roots, and the number of sign changes in $f(-x)$ is the maximum number of negative real roots.

Newton's Sums

A recursive method to compute the power sums of roots $S_k = r_1^k + r_2^k + \dots + r_n^k$ using the coefficients of a polynomial.

Sophie Germain Identity

An identity used to factor expressions of the form $a^4 + 4b^4 = (a^2 + 2b^2 - 2ab)(a^2 + 2b^2 + 2ab)$.

Circle (Locus Definition)

The set of all points in a plane that are a fixed distance $r$ (radius) from a fixed point $C$ (center).

Parabola (Locus Definition)

The set of all points that are the same distance from a fixed point (focus) and a fixed line (directrix).

Ellipse (Locus Definition)

The set of all points such that the sum of the distances from two fixed points (foci) is constant.

Hyperbola (Locus Definition)

The set of all points such that the absolute difference of the distances to two fixed points (foci) is constant.

Eccentricity ($e$)

The ratio $e = \frac{c}{a}$, which determines the type of conic section ($e=0$ for circles, $0 < e < 1$ for ellipses, $e=1$ for parabolas, and $e > 1$ for hyperbolas).

Telescoping Series

A series whose partial sums simplify due to the cancellation of intermediate terms, following the form $S_n = \sum_{i=1}^n (b_i - b_{i+1}) = b_1 - b_{n+1}$.

Circular Permutations

The number of ways to arrange $n$ distinct objects around a circle, calculated as $(n - 1)!$.

Bracelet Permutations

The number of distinct circular arrangements of $n$ objects where rotations and flips are considered identical, calculated as $\frac{(n - 1)!}{2}$.

Stars and Bars

A combinatorics method to find the number of ways to distribute $n$ indistinguishable objects into $k$ distinguishable bins, calculated as $\binom{n + k - 1}{k - 1}$.

Invertible Matrix

A square matrix $A$ that has an inverse $A^{-1}$ such that $AA^{-1} = A^{-1}A = I_n$; this occurs if and only if $\text{det}(A) \neq 0$.