Key Concepts in Polynomial and Rational Functions for College Algebra

Polynomial Functions

Definition of a Polynomial Function

A polynomial function is an expression of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$, where $n$ is a nonnegative integer and the coefficients $a_n, a_{n-1}, \ldots, a_0$ are real numbers, with $a_n \neq 0$. The degree of the polynomial is $n$, and $a_n$ is called the leading coefficient.

• Polynomial Functions: All exponents are nonnegative integers.

• Not Polynomial Functions: Exponents are not nonnegative integers or variables are under roots.

• Examples:

  • $f(x) = -3x^4 + \sqrt{2}x^2 + 5$ (Polynomial function of degree 4)

  • $g(x) = -3x^4(x - 2)(x + 3)$ (Polynomial function of degree 6)

Graphs of Polynomial Functions

Graphs of polynomial functions are characterized by smooth, rounded corners and are continuous. Non-polynomial functions may have sharp corners or discontinuities.

The Leading Coefficient Test

The leading coefficient test helps predict the end behavior of a polynomial function based on its degree and leading coefficient:

• For odd degree polynomials:

  • If leading coefficient is positive, graph falls left and rises right.

  • If leading coefficient is negative, graph rises left and falls right.

• For even degree polynomials:

  • If leading coefficient is positive, graph rises left and right.

  • If leading coefficient is negative, graph falls left and right.

Poly Behavior	Odd Degree	Even Degree
Positive Leading Coefficient	Falls to left, Rises to right	Rises to left, Rises to right
Negative Leading Coefficient	Rises to left, Falls to right	Falls to left, Falls to right

Multiplicity and x-Intercepts

If $r$ is a zero of even multiplicity, the graph touches the x-axis and turns around at $r$. If $r$ is a zero of odd multiplicity, the graph crosses the x-axis at $r$. Graphs flatten near zeros with multiplicity greater than one.

Polynomial Division and Theorems

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor $x - c$. The steps are:

1. Arrange the polynomial in descending powers, with a 0 coefficient for any missing term.

2. Write $c$ for the divisor $x - c$. To the right, write the coefficients of the dividend.

3. Write the leading coefficient of the dividend on the bottom row.

4. Multiply $c$ times the value just written on the bottom row. Write the product in the next column in the second row.

5. Add the values in this new column, writing the sum in the bottom row.

6. Repeat this series of multiplications and additions until all columns are filled in.

7. Use the numbers in the last row to write the quotient, plus the remainder above the divisor. The degree of the first term of the quotient is one less than the degree of the first term of the dividend. The final value in this row is the remainder.

The Remainder Theorem

The Remainder Theorem states: If the polynomial $f(x)$ is divided by $x - c$, then the remainder is $f(c)$.

The Factor Theorem

The Factor Theorem states:

• If $f(c) = 0$, then $x - c$ is a factor of $f(x)$.

• If $x - c$ is a factor of $f(x)$, then $f(c) = 0$.

Quadratic Functions and Equations

The Quadratic Formula

The solutions of a quadratic equation in standard form $ax^2 + bx + c = 0$, with $a \neq 0$, are given by the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The Imaginary Unit i

The imaginary unit $i$ is defined as $i = \sqrt{-1}$, where $i^2 = -1$.

Vertex of a Parabola

The vertex of a parabola defined by $f(x) = ax^2 + bx + c$ is at $\left( \frac{-b}{2a}, f\left( \frac{-b}{2a} \right) \right)$. The x-coordinate is $\frac{-b}{2a}$, and the y-coordinate is found by substituting this value into the function.

Rational Functions

Definition of Rational Functions

Rational functions are quotients of polynomial functions. They can be expressed as $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$.

Possible Rational Zeros

The possible rational zeros of a polynomial are given by:

$\text{Possible rational zeros} = \frac{\text{Factors of the constant term}}{\text{Factors of the leading coefficient}}$

Domain of Rational Functions

The domain of a rational function excludes values that make the denominator zero. For example:

• $f(x) = \frac{x^2 - 9}{x - 3}$: Domain excludes $x = 3$.

• $g(x) = \frac{x}{x^2 - 9}$: Domain excludes $x = 3$ and $x = -3$.

• $h(x) = \frac{x + 3}{x^2 + 9}$: Domain excludes values where $x^2 + 9 = 0$ (no real solutions).

Horizontal Asymptotes

A horizontal asymptote is a line $y = b$ that the graph of a function $f(x)$ approaches as $x$ increases or decreases without bound.

Locating Horizontal Asymptotes

For a rational function $f(x) = \frac{a_n x^n + \ldots + a_0}{b_m x^m + \ldots + b_0}$:

• If $n < m$, the x-axis ($y = 0$) is the horizontal asymptote.

• If $n = m$, the line $y = \frac{a_n}{b_m}$ is the horizontal asymptote.

• If $n > m$, the graph has no horizontal asymptote.

Graphing Rational Functions

Graphing rational functions involves identifying asymptotes, intercepts, and behavior near undefined points.

Applications of Quadratic Functions

Projectile Motion Example

Quadratic functions can model projectile motion, such as the height of an arrow over time. The maximum height occurs at the vertex of the parabola.

Profit Maximization Example

Quadratic functions can also model profit, where the maximum profit occurs at the vertex of the parabola.

Summary Table: Polynomial Behavior

Poly Behavior	Odd Degree	Even Degree
Positive Leading Coefficient	Falls to left, Rises to right	Rises to left, Rises to right
Negative Leading Coefficient	Rises to left, Falls to right	Falls to left, Falls to right

Additional info:

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