Complete Study Guide Flashcards: Limits, Counting, Binomial Theorem, Polynomials, Division & Inequalities

Vocabulary flashcards covering limits, asymptotes, counting principles, permutations, combinations, the binomial theorem, polynomial functions, graphing, division algorithms, and polynomial inequalities.

Limit

The mathematical concept asking what value the output $y$ gets closer to as the input $x$ approaches a specific number.

Left-Hand Limit

The height a function approaches as $x$ approaches a number $a$ strictly from values less than $a$ (from the left), denoted $\lim_{x \to a^-} f(x)$.

Right-Hand Limit

The height a function approaches as $x$ approaches a number $a$ strictly from values greater than $a$ (from the right), denoted $\lim_{x \to a^+} f(x)$.

Two-Sided Limit Existence

The condition that a two-sided limit $\lim_{x \to a} f(x)$ exists if and only if the left-hand limit equals the right-hand limit: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$.

Jump Discontinuity

A feature on a graph where the left-hand and right-hand limits approach different finite numbers, causing the two-sided limit to not exist (DNE).

Open Circle

A point on a graph indicating that the function is not defined at that exact spot, representing a hole in the curve.

Closed Dot

A filled-in point on a graph indicating the actual value of the function $f(a)$ plotted at $x = a$.

Limit at Infinity

A limit of the form $\lim_{x \to \infty} f(x)$ or $\lim_{x \to -\infty} f(x)$ describing the horizontal height a graph settles toward as $x$ moves far to the right or left.

Infinite Limit

A limit of the form $\lim_{x \to c} f(x) = \infty$ or $-\infty$ where the graph shoots straight up or down without bound as $x$ approaches a finite number $c$.

Horizontal Asymptote

A horizontal line $y = L$ that the graph flattens toward as $x \to \infty$ or $x \to -\infty$.

Vertical Asymptote

A vertical line $x = c$ along which a graph shoots upward toward $\infty$ or downward toward $-\infty$ as $x$ approaches $c$.

Fundamental Counting Principle

The rule stating that if event 1 can happen in $m$ ways and event 2 in $n$ ways, then both happening in order occur in $m \times n$ ways.

Factorial

The product of all whole numbers from $n$ down to 1, denoted $n! = n \times (n - 1) \times \dots \times 1$, with $0! = 1$ by convention.

Permutation

An arrangement of items where the specific order of selection changes the outcome.

Permutation Formula

The formula $P(n, r) = \frac{n!}{(n - r)!}$ used to compute the number of ways to choose and arrange $r$ items out of $n$ distinct items.

Distinguishable Permutations

The number of unique arrangements of $n$ objects when some objects are identical copies, computed as $\frac{n!}{n_1! \, n_2! \, \dots \, n_k!}$.

Combination

A selection or group of items where the order of selection does not matter.

Combination Formula

The formula $C(n, r) = \frac{n!}{r! \, (n - r)!}$ used to compute the number of ways to pick $r$ items from $n$ distinct items without regard to order.

Total Subsets Formula

The rule stating that a set with $n$ elements has a total of $2^n$ possible subsets, including the empty set and the full set.

Pascal's Triangle

A triangular array of numbers where each interior entry is the sum of the two numbers diagonally above it, representing the coefficients of $(a + b)^n$.

Binomial Coefficient

A formula given by $\binom{n}{r} = \frac{n!}{r! \, (n - r)!}$, which matches $C(n, r)$ and calculates specific entries in Pascal's Triangle.

Binomial Theorem

The formula $(a + b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^r$ used to expand a binomial raised to any nonnegative integer power $n$.

General Term of a Binomial Expansion

The single term containing $a^r$ in the expansion of $(a + b)^n$, calculated directly using $\binom{n}{n - r} a^r b^{n - r}$.

Polynomial Function

A function formed by nonnegative whole-number powers of $x$ multiplied by constant coefficients and added together: $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$.

Degree of a Polynomial

The highest exponent $n$ among all terms in a polynomial.

Leading Term

The term $a_n x^n$ containing the highest degree in a polynomial, where $a_n$ is called the leading coefficient.

End Behavior

The behavior of a polynomial $P(x)$ as $x \to \infty$ or $x \to -\infty$, which is entirely dictated by its leading term $a_n x^n$.

Root of a Polynomial

An $x$-value $a$ such that $P(a) = 0$, representing a location where the graph touches or crosses the $x$-axis.

Multiplicity

The exponent $n$ associated with a factor $(x - a)^n$ in a completely factored polynomial.

Odd Multiplicity Root

A root whose factor exponent is odd (1, 3, 5, …), causing the polynomial graph to pass straight through the $x$-axis.

Even Multiplicity Root

A root whose factor exponent is even (2, 4, 6, …), causing the polynomial graph to touch the $x$-axis and bounce off.

Relative Extrema Limit

The rule stating that a polynomial of degree $n$ can have at most $n - 1$ relative extrema (turning points).

Division Algorithm for Polynomials

The principle stating that dividing $P(x)$ by $d(x)$ yields quotient $q(x)$ and remainder $r(x)$ such that $P(x) = q(x) \cdot d(x) + r(x)$ with $0 \le \text{deg}(r) < \text{deg}(d)$.

Synthetic Division

A streamlined shortcut for dividing polynomials that uses only coefficients and is strictly valid for linear divisors of the form $x - c$.

Polynomial Inequality

An expression comparing a polynomial to zero, solved by finding roots and determining the sign of the polynomial in each sub-interval.

Test-Point Method

A technique for solving polynomial inequalities by substituting a sample number from each number-line interval into the polynomial to check its sign.

End-Behavior and Multiplicity Shortcut

A method for solving polynomial inequalities by establishing the sign of the rightmost interval via end behavior and toggling signs across odd multiplicity roots while retaining signs across even multiplicity roots.

Example 1 Polynomial Inequality Solution

The solution to the inequality $x^2 - x - 6 > 0$, expressed as $x < -2 \text{ or } x > 3$.

Example 2 Polynomial Inequality Solution

The solution to the inequality $(x - 1)^2(x + 3) \le 0$, expressed as $x \le -3 \text{ or } x = 1$.

Example 3 Polynomial Inequality Solution

The solution to the inequality $(x + 2)(x - 1)(x - 4) < 0$, expressed as $x < -2 \text{ or } 1 < x < 4$.