Mathematics Volume 1: Algebra Lecture Notes

Practice flashcards covering algebraic terminologies, exponent laws, special products, factoring techniques, function types, asymptotes, logarithms, and set notation based on the Algebra Volume 1 lecture.

What is an algebraic expression?

A combination of numbers and letters connected by arithmetic operations.

What is the difference between a constant and a variable?

A constant is a number that does not change (e.g., $6$, $4$, and $10$), while a variable is a symbol, usually a letter, that changes value depending on the problem (e.g., $x$ and $y$).

What are 'Like Terms' in algebra?

Terms having the same variables wherein addition and subtraction are possible.

What is the FOIL Method used for?

A technique used for multiplying binomials; it stands for "First, Outside, Inside, Last."

According to the Laws of Exponents, what is the result of $(x^a)(x^b)$?

$$x^{(a+b)}$$

What is the formula for a fractional exponent $x^{a/b}$?

$\sqrt[b]{x^a}$ or $(\sqrt[b]{x})^a$

What is the Quadratic Formula for solving unknowns when $ax^2 + bx + c = 0$?

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

What is the expanded form of the special product $(a \pm b)^3$?

$$a^3 \pm 3a^2b + 3ab^2 \pm b^3$$

What is the expanded form of the trinomial square $(a + b + c)^2$?

$$a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$$

In factoring, what does the AC Method involve?

Finding two numbers whose product is $a \times c$ and whose sum is $b$ when $a \neq 1$ in the form $ax^2 + bx + c$.

What is the Vertical Line Test?

A test where if a vertical line crosses a curve only once, the curve is a function; otherwise, it is not a function.

In the context of functions, define Domain and Range.

Domain refers to the possible inputs for a function ($x$), while Range refers to all possible outputs of a function ($f(x)$).

What is the Intercept Form of a linear equation?

$\frac{x}{a} + \frac{y}{b} = 1$, where $a$ is the x-intercept and $b$ is the y-intercept.

According to the lecture notes, what is the 'Point-Slope' form of a Quadratic Function?

$y = a^2(x - h)^2 + k$, where $a$ is the point of reflection, $h$ is the x-coordinate vertex, and $k$ is the y-coordinate vertex.

How is a Horizontal Asymptote (HA) determined for a rational function?

By comparing the degrees of the numerator ($N$) and denominator ($D$): If $N < D$, HA is $y = 0$; if $N > D$, there is no HA; if $N = D$, then $y = N/D$ (ratio of coefficients).

What is a Composite Function?

Functions within a function, denoted as $f[g(x)]$ or $(f \circ g)(x)$, solved by substituting the inner function into the outer function.

What is the difference between Common Logarithms and Natural Logarithms?

Common Logarithms have a base of $10$ ($\log_{10} a$), while Natural Logarithms have a base of $e = 2.7182828$ ($\ln a$).

What rule applies when solving inequalities involving multiplication or division by negative signs?

Multiplying or dividing values with negative signs reverses the inequality (e.g., $>$ becomes $<$).

How are the Roster and Set Builder methods of set notation defined?

Roster lists all elements explicitly within braces (e.g., $A = \{12, 14, 16, 18\}$), while Set Builder uses a general property form (e.g., $\{x | properties\}$).

What does the Power Set $P(C)$ represent?

The collection of all subsets of set $C$, given by the formula $2^s$, where $s$ is the number of elements in $C$.

What is the 'Symmetric Difference' ($A \Delta B$) of two sets?

The set of objects that belong to $A$ or $B$ but not to their intersection.

Define the Integer Numbers Set ($Z$) and the Rational Numbers Set ($Q$).

$Z = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$ and $Q = \{x | x = a/b, a, b \in Z \}$.