BCPC 105: Introduction to Business Mathematics

Comprehensive vocabulary flashcards covering the core algebraic, financial, and analytical mathematics concepts from the BCPC 105 course notes.

Polynomial

An algebraic expression of the form $a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \dots + a_1 x + a_0$, where $a_0, a_1, a_2, \dots, a_n$ are real numbers, $n$ is a natural number, and $a_n \neq 0$.

Prime Polynomial

A polynomial that cannot be factored into factors of lower degree.

Greatest Common Factor (Polynomial)

The largest factor or quantity that divides evenly into every term of a polynomial expression.

Linear Equation

An algebraic equation that can be written in the form $ax + b = 0$, where $a$ and $b$ are real numbers and $a \neq 0$.

Parallel Lines

Two non-vertical lines that have identical slopes or gradients.

Perpendicular Lines

Two lines, neither of which is vertical, whose slopes or gradients have a product of $-1$.

Linear Cost Function

A total cost model expressed as $C(x) = mx + b$, where $m$ denotes the marginal cost per item and $b$ denotes the fixed cost.

Quadratic Function

A function defined by $f(x) = ax^2 + bx + c$, where $a, b,$ and $c$ are real numbers and $a \neq 0$, whose graph is concave if $a < 0$ and convex if $a > 0$.

Remainder Theorem

The principle stating that when a polynomial $f(x)$ is divided by $(x - a)$, the remainder obtained is equal to $f(a)$, and when divided by $(x + a)$, the remainder is $f(-a)$.

Factor Theorem

The theorem stating that if $f(a) = 0$ for a polynomial $f(x)$, then $(x - a)$ is a factor of $f(x)$.

Cubic Equation

A polynomial equation of degree three written in the form $ax^3 + bx^2 + cx + d = 0$, where $a, b, c,$ and $d$ are constants and $a \neq 0$.

Linear Demand Function

An economic function expressing quantity demanded or unit price with a negative slope, written in the form $D(p) = mp + b$ or $P = mQ + b$, where $m < 0$.

Linear Supply Function

An economic function expressing quantity supplied or unit price with a positive slope, written in the form $S(p) = mp + b$ or $P = mQ + b$, where $m > 0$.

Market Equilibrium

The point where the quantity demanded equals the quantity supplied ($D(Q) = S(Q)$), establishing the equilibrium price and equilibrium quantity.

Break-Even Point

The sales level or quantity where a firm makes neither a profit nor a loss ($\pi(Q) = 0$ or $R(Q) = C(Q)$), fully covering all total costs.

Matrix

A rectangular array of numbers arranged into $m$ rows and $n$ columns, designated by the dimension or order $(m \times n)$.

Equal Matrices

Two matrices that have the exact same dimension or order and whose elements in corresponding positions are identical.

Square Matrix

A matrix in which the number of rows is equal to the number of columns ($m = n$).

Unit (Identity) Matrix

A square matrix with ones along the principal diagonal and zeros in every other position, denoted by $I$.

Transpose of a Matrix

A new matrix, denoted by $A^T$ or $A'$, formed by interchanging the rows and columns of matrix $A$, such that the $i\text{th}$ row becomes the $j\text{th}$ column.

Symmetric Matrix

A square matrix $A$ that is equal to its transpose ($A = A^T$).

Determinant

A scalar numerical value assigned to a square matrix, denoted by $\det(A)$ or $|A|$.

Adjoint of a Matrix

The matrix formed by taking the transpose of the cofactor matrix of a given square matrix.

Cramer's Rule

A method for solving systems of linear equations using determinants, in which each unknown variable is solved by dividing the determinant of the modified variable matrix by the determinant of the coefficient matrix.

Common Logarithm

A logarithm evaluated to base $10$, written as $\log_{10}$, $\lg_{10}$, or simply $\log$.

Natural Logarithm

A logarithm evaluated to base $e$, written as $\ln$, where $e$ is the mathematical exponential constant.

Surd

An irrational root of the form $\sqrt[n]{a}$, representing an irrational number of the $n\text{th}$ order.

Conjugate Surds

A pair of radical binomial expressions differing only in the sign connecting their terms, such as $a + \sqrt{b}$ and $a - \sqrt{b}$.

Cash Discount

A percentage reduction subtracted from the marked or catalogued price of goods when purchased and settled immediately in cash.

Profit Margin

Profit expressed as a percentage of the selling price (sales revenue), in which the selling price represents $100\%$.

Mark-up

Profit expressed as a percentage of the cost of sales, representing the amount added to the cost price to determine the final selling price.

Commission

A service fee paid to a buying or selling agent, typically calculated as a percentage of the total sales or purchase volume.

Proportion

A percentage expressed as a decimal fraction of $1$, where $100\%$ equals $1$.

Ratio

A mathematical statement indicating the relative share or division of components comprising a whole.

Rational Function

A mathematical function expressed as the ratio of two polynomials, $f(x) = \frac{p(x)}{q(x)}$, where $q(x) \neq 0$.

Ordinary Interest vs. Exact Interest

Ordinary interest is simple interest computed using a $360$-day calendar year, while exact interest is computed using a $365$-day calendar year.

Compound Interest

An interest accumulation system in which interest earned across each period is added to the principal to generate interest in subsequent periods.

Effective Rate

The actual annual percentage yield $r_e$ produced by a stated nominal interest rate $r$ compounded $m$ times per year, given by $r_e = \left(1 + \frac{r}{m}\right)^m - 1$.

Ordinary Annuity

A financial sequence of equal periodic payments or receipts occurring at the end of each compounding period, where payment frequency matches compounding frequency.

Amortization

The process of retiring a debt through a series of equal periodic payments that cover both the principal balance and accrued interest.

Straight-Line Method of Depreciation

A linear depreciation model assuming a uniform value loss each year, calculated by dividing the total depreciable base by useful life: $d = \frac{A_0 - A_t}{n}$, where $A_0$ is original cost, $A_t$ is scrap value, and $n$ is years.