College Algebra: Polynomials, Exponent Rules, Factoring, Rational Expressions, and Graphs

Vocabulary practice flashcards covering fundamental College Algebra concepts from MATH 1111: polynomials (P.4), exponent rules (P.2 review), factoring polynomials (P.5), rational expressions (P.6), and Cartesian graphing (1.1).

Monomial

An expression of the type $ax^n$, where $a$ is a real-number constant and $n$ is a nonnegative integer; an algebraic expression consisting of exactly $1$ term.

Binomial

A polynomial that contains exactly $2$ terms.

Trinomial

A polynomial that contains exactly $3$ terms.

Polynomial

A monomial or a combination of sums and/or differences of monomials; 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 $a_n, \dots, a_0$ are real numbers, $a_n \neq 0$, and $n$ is a nonnegative integer.

Non-Polynomial Expression Condition

If a variable appears in the denominator (or with a negative exponent), the expression is not a polynomial.

Terms of a Polynomial

The monomials together with their sign; mathematical expressions that are added or subtracted, always separated by plus or minus signs.

Coefficients

The numbers in front of the monomials together with their sign that indicate 'how many'. They can be whole numbers, negatives, fractions, decimals, and/or roots.

Standard Form of a Polynomial

The arrangement of a polynomial where the terms are written in descending order of their exponents.

Degree of a Term

The exponent of the variable in that monomial term. The degree of any nonzero constant term is $0$.

Degree of a Polynomial

The greatest degree among all of the individual terms comprising the polynomial.

Leading Coefficient

The coefficient of the term of highest degree ($a_n$) in a polynomial.

Constant Term

The term in a polynomial that contains no variable ($a_0$), having a degree of $0$.

Like Terms

Terms that contain the identical variable raised to the same power.

Factors

Mathematical expressions that are multiplied (or divided), separated by multiplication or division symbols.

Distributive Property

The algebraic property stating $c(a + b) \Rightarrow ca + cb$, which converts factors (multiplication) into terms (addition) by removing parentheses.

Base

In exponential notation $a^n$, the value $a$ that is multiplied repeatedly.

Exponent

In exponential notation $a^n$, the value $n$ indicating how many times the base is used as a factor.

Exponential Notation

An expression written as $a^n$, signifying $a \cdot a \cdot a \cdots a$ multiplied $n$ times; read as '$a$ to the $n^{\text{th}}$ power'.

Exponents of 0 and 1 Rules

Rules stating that $a^1 = a$ for any number $a$, and $a^0 = 1$ for any nonzero number $a$.

Negative Exponent Rule

For any nonzero real number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$.

The Product Rule for Exponents

For any number $a$ and positive integers $m$ and $n$, $a^m \cdot a^n = a^{m+n}$; when multiplying expressions with like bases, keep the base and add the exponents.

The Quotient Rule for Exponents

For any nonzero number $a$ and positive integers $m$ and $n$, $\frac{a^m}{a^n} = a^{m-n}$; when dividing expressions with like bases, keep the base and subtract the denominator's exponent from the numerator's exponent.

The Power Rule for Exponents

For any real number $a$ and integers $m$ and $n$, $(a^m)^n = a^{mn}$; to raise a power to a power, multiply the exponents.

Raising a Product to a Power Rule

For any real numbers $a$ and $b$ and any integer $n$, $(ab)^n = a^n b^n$; to raise a product to the $n^{\text{th}}$ power, raise each factor to the $n^{\text{th}}$ power.

Raising a Quotient to a Power Rule

For any real numbers $a$ and $b$ ($b \neq 0$) and integer $n$, $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, and $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n = \frac{b^n}{a^n}$ ($a \neq 0$).

Factoring

The process of un-distributing, transforming terms (addition/subtraction) into factors (multiplication) by adding parentheses ($ab + ac = a(b + c)$).

Greatest Common Factor (GCF)

The largest factor common to all terms in a polynomial that can be factored out.

Binomial Factor

A two-term common factor shared across terms in an expression, such as $(x - 2)$ in $x^2(x - 2) + 3(x - 2) = (x - 2)(x^2 + 3)$.

Factoring by Grouping

A method used to factor four-term polynomials by pairing terms into two groups, factoring the GCF out of each group, and then factoring out the shared binomial factor.

Prime Polynomial

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

The ac-Method

A method used to factor trinomials of the form $ax^2 + bx + c$ ($a \neq 1$) by multiplying $a$ and $c$, finding integers $p$ and $q$ such that $pq = ac$ and $p + q = b$, splitting the middle term, and factoring by grouping.

Difference of Squares

A two-term polynomial of the format $A^2 - B^2$, which factors completely into $(A + B)(A - B)$.

Perfect Square Trinomial

A trinomial resulting from squaring a binomial, factoring into the square of a single binomial (for example, $25x^2 + 10x + 1 = (5x + 1)^2$).

Rational Expression

An algebraic expression representing the quotient (ratio, fraction) of two polynomials.

Domain of an Algebraic Expression

The set of all real numbers for which an algebraic expression is defined.

Restricted Values

Numbers that make the denominator of a rational expression equal to $0$ and therefore must be excluded from its domain.

Dividing Rational Expressions Rule

The rule stating that two rational expressions are divided by multiplying the dividend by the reciprocal of the divisor: $\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}$.

Rectangular Coordinate System

A two-dimensional plane formed by two perpendicular number lines (the $x$-axis and $y$-axis) that intersect at the origin.

x-axis

The horizontal number line in a rectangular coordinate system.

y-axis

The vertical number line in a rectangular coordinate system.

Origin

The point of intersection of the $x$-axis and $y$-axis on a rectangular coordinate system, represented by the ordered pair $(0, 0)$.

Quadrants

The four regions into which the coordinate plane is divided by the horizontal and vertical axes, labeled Q1, Q2, Q3, and Q4.

Ordered Pair

A pair of numbers $(x, y)$ representing the location of a point; structured as $(horizontal, vertical)$, $(x\text{-value}, y\text{-value})$, or $(input, output)$.

x-coordinate

The first number in an ordered pair $(x, y)$, denoting the horizontal directed distance of the point from the $y$-axis.

y-coordinate

The second number in an ordered pair $(x, y)$, denoting the vertical directed distance of the point from the $x$-axis.

Solution to an Equation in Two Variables

An ordered pair $(x, y)$ that makes an equation in two variables a true statement when substituted.

Graph of an Equation

The set of all points on the rectangular coordinate plane whose coordinates satisfy the given equation.

Point-Plotting Method

A method of graphing an equation by choosing several $x$-values, calculating the corresponding $y$-values in a table, plotting those ordered pairs, and connecting them.

x-intercept

A point where a graph crosses or touches the $x$-axis; found algebraically by setting $y = 0$ and solving for $x$.

y-intercept

A point where a graph crosses or touches the $y$-axis; found algebraically by setting $x = 0$ and solving for $y$.