Polynomial and Rational Functions Review

Vocabulary practice cards covering rational functions, polynomial forms, area model division, slant asymptotes, Pascal's triangle binomial expansion, and end behavior limits.

x-intercept of a Rational Function

An input value $x = a$ where the numerator is zero ($0$) and the denominator is non-zero, resulting in $f(a) = \frac{0}{\text{non-0}} = 0$.

Hole (Removable Discontinuity)

A point where both the numerator and denominator of a rational function equal zero ($\frac{0}{0}$); its $y$-value is found by substituting the $x$-value into the simplified version of the function.

Vertical Asymptote

A vertical line $x = c$ where the denominator equals zero while the numerator is non-zero ($\frac{\text{non-0}}{0}$), causing function outputs to increase or decrease without bound near $x = c$.

Factor of a Polynomial

An expression, such as $x + 1$, that divides into a polynomial $g(x)$ completely with a remainder of $0$.

Multiplicity

The number of times a given factor or root appears in the factored form of a polynomial function, such as a root occurring twice having a multiplicity of $2$.

Area Model (Polynomial Division)

A box-grid method used to divide polynomials, determine unknown factors, and identify the remainder of a division problem.

Mixed Form of a Rational Function

A rational function rewritten as the sum of a polynomial quotient and a proper rational remainder expression, in the form $f(x) = q(x) + \frac{r(x)}{d(x)}$.

Slant (Oblique) Asymptote

A linear asymptote of the form $y = q(x)$ that occurs when the degree of the numerator polynomial is exactly $1$ greater than the degree of the denominator polynomial.

Rational Function Division Form

The representation $\frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}$, where $f(x)$ is the dividend, $d(x)$ is the divisor, $q(x)$ is the quotient, and $r(x)$ is the remainder.

Pascal's Triangle

A triangular array of numbers where each row $n$ provides the binomial coefficients for expanding an expression of the form $(a + b)^n$.

Row n Property of Pascal's Triangle

The structural property stating that the $n^{\text{th}}$ row of Pascal's Triangle always contains exactly $n + 1$ terms.

Three-Column Method for Binomial Expansion

An expansion organizer where Column 1 contains row $n$ of Pascal's Triangle, Column 2 contains descending powers from $a^n$ to $a^0$, and Column 3 contains ascending powers from $b^0$ to $b^n$.

End Behavior (Rational Functions)

The limiting value or graphical trajectory of $f(x)$ as $x \rightarrow \infty$ and $x \rightarrow -\infty$, determined by the simplified ratio of the leading terms of the numerator and denominator.

End Behavior Rule: Greater Degree in Denominator

When the degree of the denominator is greater than the degree of the numerator ($m > n$), the end behavior approaches zero: $\lim_{x \to \pm \infty} f(x) = 0$.

End Behavior Rule: Equal Degrees

When the degree of the numerator equals the degree of the denominator ($n = m$) for leading terms $\frac{a x^n}{b x^m}$, the end behavior is the ratio of their leading coefficients: $\lim_{x \to \pm \infty} f(x) = \frac{a}{b}$.

End Behavior Rule: Greater Degree in Numerator

When the degree of the numerator is greater than the degree of the denominator ($n > m$), the end behavior approaches $\pm \infty$, modeling the shape of a polynomial curve or line.