Polynomial and Rational Functions: Zeros, Factorization, and Graph Behavior

Polynomial Functions

Zeros of Polynomial Functions

Finding the zeros (roots) of a polynomial is a fundamental concept in College Algebra. Zeros are the values of x for which the polynomial equals zero.

• Definition: A zero of a polynomial function f(x) is a value r such that f(r) = 0.

• Example: For f(x) = x^2 - 4x + 5, set x^2 - 4x + 5 = 0 and solve for x using the quadratic formula:

$x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 20}}{2} = \frac{4 \pm \sqrt{-4}}{2} = 2 \pm i$

• Rational Zeros Theorem: For a polynomial with integer coefficients, possible rational zeros are of the form $\pm \frac{\text{factors of constant term}}{\text{factors of leading coefficient}}$.

• Example: For f(x) = x^3 - x^2 - 37x + 35, possible rational zeros are $\pm 1, \pm 35, \pm 7, \pm 5$.

Constructing Polynomials from Zeros

Given zeros, a polynomial can be constructed by multiplying factors corresponding to each zero.

• General Form: If zeros are $r_1, r_2, \ldots, r_n$, then $f(x) = a(x - r_1)(x - r_2)\ldots(x - r_n)$ where a is the leading coefficient.

• Example: Zeros at $-3, 3, 9$ yield $f(x) = (x + 3)(x - 3)(x - 9)$.

Factoring Polynomials

Factoring expresses a polynomial as a product of lower-degree polynomials, often linear or quadratic factors.

• Linear Factors: Each zero corresponds to a linear factor $(x - r)$.

• Multiplicity: If a zero has multiplicity $k$, its factor is raised to the $k$th power.

• Example: For $f(x) = x^3 - 7x^2 + 25x - 39$ with zeros at $x = 5, x = -4, x = 2$, the factorization is $(x - 5)(x + 4)(x - 2)$.

End Behavior of Polynomial Functions

The end behavior describes how the function behaves as $x \to \infty$ or $x \to -\infty$. It is determined by the leading term.

• Even Degree, Positive Leading Coefficient: Both ends up.

• Even Degree, Negative Leading Coefficient: Both ends down.

• Odd Degree, Positive Leading Coefficient: Left end down, right end up.

• Odd Degree, Negative Leading Coefficient: Left end up, right end down.

• Example: $f(x) = 5x^4 - 3x^2 + 9$ (even degree, positive coefficient) → both ends up.

Zeros and Sign Changes: Descartes' Rule of Signs

Descartes' Rule of Signs

This rule helps estimate the number of positive and negative real zeros of a polynomial.

• Positive Real Zeros: Count sign changes in coefficients of $f(x)$.

• Negative Real Zeros: Substitute $x$ with $-x$ in $f(x)$ and count sign changes.

• Example: For $f(x) = x^3 + x^2 - x - 2$, sign changes indicate possible positive zeros.

Intermediate Value Theorem

If $f(a)$ and $f(b)$ have opposite signs, there is at least one zero between $a$ and $b$.

• Example: If $f(6) = -4$ and $f(5) = 7$, there is a zero between $x = 5$ and $x = 6$.

Graphing Polynomial Functions

Intercepts and Turning Points

Intercepts are points where the graph crosses the axes. Turning points are local maxima or minima.

• x-intercepts: Set $f(x) = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $f(0)$.

• Turning Points: A polynomial of degree $n$ has at most $n-1$ turning points.

Rational Functions

Properties of Rational Functions

A rational function is a ratio of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$. Key features include domain, intercepts, and asymptotes.

• Domain: All real numbers except where $Q(x) = 0$.

• x-intercepts: Values of $x$ where $P(x) = 0$.

• y-intercept: $f(0)$.

• Vertical Asymptotes: Values of $x$ where $Q(x) = 0$ and not canceled by $P(x)$.

• Horizontal Asymptotes: Determined by degrees of $P(x)$ and $Q(x)$.

• Slant Asymptotes: Occur when degree of $P(x)$ is one more than degree of $Q(x)$.

• Holes: Occur at values canceled from both numerator and denominator.

Example Table: Rational Function Features

Feature	How to Find	Example
Domain	Set denominator ≠ 0	$x \neq 3, x \neq -3$
x-intercept	Set numerator = 0	$x = 0$
y-intercept	Evaluate $f(0)$	$f(0) = 14$
Vertical Asymptote	Set denominator = 0	$x = 3, x = -3$
Horizontal Asymptote	Compare degrees	$y = 9$
Slant Asymptote	Degree numerator = degree denominator + 1	None

Graphing Rational Functions

To graph a rational function, plot intercepts, asymptotes, and holes, then sketch the general shape.

• Example: For $f(x) = \frac{(x+7)(x-2)}{(x-3)(x+3)}$, vertical asymptotes at $x = 3, x = -3$, hole at $x = 7$.

Vertex Form and Completing the Square

Completing the Square

Completing the square rewrites a quadratic in vertex form $f(x) = a(x-h)^2 + k$, revealing the vertex $(h, k)$.

• Steps:

  1. Group $x^2$ and $x$ terms.

  2. Add and subtract $(\frac{b}{2a})^2$ inside the equation.

  3. Rewrite as $a(x-h)^2 + k$.

• Example: $x^2 - 6x + 18 = (x-3)^2 + 9$; vertex at $(3, 9)$.

Summary Table: Polynomial and Rational Function Concepts

Concept	Key Points	Example
Zeros	Set $f(x) = 0$	$x = 2 \pm i$
Rational Zeros	Possible zeros: $\pm \frac{\text{factors of constant}}{\text{factors of leading coefficient}}$	$\pm 1, \pm 35, \pm 7, \pm 5$
Factoring	Express as product of linear/quadratic factors	$(x-5)(x+4)(x-2)$
End Behavior	Determined by leading term	Even degree, positive coefficient: both ends up
Descartes' Rule	Sign changes = possible positive/negative zeros	Alternate signs: no zero greater than 2
Vertex Form	Complete the square	$(x-3)^2 + 9$

Additional info: Some explanations and examples have been expanded for clarity and completeness beyond the original handwritten notes.