Polynomial and Rational Functions Flashcards

Flashcards covering key vocabulary, theorems, and definitions for Power, Polynomial, Quadratic, and Rational functions as detailed in Chapter 3 of Precalculus: An Investigation of Functions.

Power Function

A function that can be represented in the form $f(x) = x^p$ where the base is a variable and the exponent, $p$, is a number.

Polynomial

The sum of terms each consisting of a vertically stretched or compressed power function with non-negative whole number power.

Degree

The highest power of the variable that occurs in the polynomial.

Leading Term

The term containing the highest power of the variable: the term with the highest degree.

Leading Coefficient

The coefficient of the leading term.

Long Run Behavior

The behavior of the graph of a function as the input takes on large negative values, $x \rightarrow -\infty$, and large positive values, $x \rightarrow \infty$.

Short Run Behavior

Characteristics of the graph such as vertical and horizontal intercepts and the places the graph changes direction.

Turning Point

A place where the graph of a polynomial changes direction.

Vertex

The specific feature of a quadratic graph where the function changes direction; the horizontal and vertical shifts of the basic quadratic determine its location.

Standard Form (Quadratic)

A quadratic function written in the form $f(x) = ax^2 + bx + c$.

Transformation Form / Vertex Form

A quadratic function written in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

Completing the Square

A technical process used to rewrite a quadratic function from standard form into vertex form by creating a perfect square trinomial.

Quadratic Formula

The formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ used to find the horizontal intercepts of a quadratic function in standard form.

Multiplicity

The power $p$ on a factor $(x - h)^p$ which determines the graphical behavior at a horizontal intercept; also known as the number of times a zero is repeated.

Remainder Theorem

If $p(x)$ is a polynomial of degree 1 or greater and $c$ is a real number, then when $p(x)$ is divided by $x - c$, the remainder is $p(c)$.

Factor Theorem

States that a real number $c$ is a zero of a nonzero polynomial $f$ if and only if $x - c$ is a factor of $f(x)$.

Synthetic Division

A streamlined tool for dividing polynomials by divisors of the form $x - c$, using only the coefficients of the terms.

Cauchy's Bound

Determines an interval $[-\frac{M}{|a_n|} - 1, \frac{M}{|a_n|} + 1]$ containing all real zeros of a polynomial, where $M$ is the largest coefficient in absolute value.

Rational Roots Theorem

States that if a polynomial with integer coefficients has a rational zero $r$, it must be of the form $r = \pm \frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

Imaginary Number $i$

Defined as $i = \sqrt{-1}$.

Complex Number

A number in the form $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Complex Conjugate

The conjugate of a complex number $a + bi$ is $a - bi$; multiplying a complex number by its conjugate results in a real number.

Fundamental Theorem of Algebra

A non-constant polynomial $f$ with real or complex coefficients will have at least one real or complex zero.

Complex Factorization Theorem

States that if $f$ is a polynomial of degree $n \geq 1$, then $f$ has exactly $n$ real or complex zeros, counting multiplicities.

Rational Function

A function that can be written as the ratio of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$.

Vertical Asymptote

A vertical line $x = a$ where the graph tends toward positive or negative infinity as the inputs approach $a$.

Horizontal Asymptote

A horizontal line $y = b$ where the graph approaches the line as the inputs get large ($x \rightarrow \pm \infty$).

Hole

A single point where the graph of a rational function is not defined, occurring when both the numerator and denominator are zero at the same input.

Oblique Asymptote

The asymptotic behavior of a rational function where the long-run behavior follows a linear quotient obtained from polynomial long division; also called a slant asymptote.

Radical Function

Functions involving roots, such as square or cube roots, often arising as the inverses of polynomial functions on restricted domains.