Polynomial and Rational Functions: Precalculus Study Guide

Polynomial Functions

Definition and Classification

Polynomial functions are algebraic expressions consisting of variables and coefficients, involving only non-negative integer powers of the variable. They are fundamental in precalculus and are used to model various real-world phenomena.

• Polynomial of degree n: An expression of the form $a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_1x + a_0$, where $n$ is a non-negative integer and $a_n eq 0$.

• Non-polynomial expressions: If the variable is raised to a negative, fractional, or non-integer power, or if the constant term is absent, the expression is not a polynomial.

• Standard form: Write the polynomial in descending order of powers, identifying the leading term (highest degree term) and constant term (term without variable).

Zeros and Multiplicity

The zeros (roots) of a polynomial are the values of $x$ for which the polynomial equals zero. The multiplicity of a zero refers to how many times a particular root occurs.

• Multiplicity: If $(x - r)^k$ is a factor, $r$ is a zero of multiplicity $k$.

• Graphical behavior: If the multiplicity is odd, the graph crosses the x-axis at the zero; if even, it touches but does not cross.

• Example: For $f(x) = (x-2)^2(x+1)$, $x=2$ has multiplicity 2, $x=-1$ has multiplicity 1.

Degree and Turning Points

The degree of a polynomial determines its general shape and the maximum number of turning points.

• Degree: The highest power of $x$ in the polynomial.

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

• Example: A cubic polynomial ($n=3$) can have up to 2 turning points.

Constructing Polynomials from Graphs

Given a graph, you can construct a polynomial by identifying its zeros, their multiplicities, and the end behavior.

• End behavior: Determined by the leading term. For $f(x) = ax^n$, if $n$ is even and $a > 0$, both ends rise; if $a < 0$, both ends fall. If $n$ is odd, ends go in opposite directions.

• Example: If a graph crosses the x-axis at $x=1$ and $x=3$, and touches at $x=-2$, a possible polynomial is $f(x) = a(x-1)(x-3)(x+2)^2$.

Polynomial Theorems and Zeros

Rational Zeros Theorem

This theorem helps to find all possible rational zeros of a polynomial function.

• Possible rational zeros: If $f(x) = a_nx^n + ... + a_0$, possible rational zeros are $rac{p}{q}$, where $p$ divides $a_0$ and $q$ divides $a_n$.

• Example: For $f(x) = 2x^3 - 4x^2 - 5x + 6$, possible rational zeros are $rac{ ext{factors of }6}{ ext{factors of }2}$.

Intermediate Value Theorem

This theorem states that if a polynomial function changes sign over an interval, it must have a zero in that interval.

• Application: If $f(a)$ and $f(b)$ have opposite signs, there exists $c$ in $(a, b)$ such that $f(c) = 0$.

Remainder and Factor Theorems

These theorems are used to evaluate polynomials and determine factors.

• Remainder Theorem: The remainder of $f(x)$ divided by $(x - k)$ is $f(k)$.

• Factor Theorem: $(x - k)$ is a factor of $f(x)$ if and only if $f(k) = 0$.

Rational Functions

Definition and Domain

A rational function is a ratio of two polynomials, $R(x) = rac{P(x)}{Q(x)}$, where $Q(x) eq 0$.

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

• Example: For $H(x) = rac{x^2}{(x+1)(x-4)}$, domain is $(-, -1)  (-1, 4)  (4, )$.

Intercepts

Intercepts are points where the graph crosses the axes.

• x-intercepts: Set numerator equal to zero, solve for $x$.

• y-intercept: Set $x=0$, solve for $y$.

Asymptotes

Asymptotes are lines that the graph approaches but never touches.

• Vertical asymptotes: Set denominator equal to zero, solve for $x$.

• Horizontal asymptotes: Compare degrees of numerator and denominator:

  • If degree numerator < degree denominator: $y=0$

  • If degrees equal: $y = rac{ ext{leading coefficient of numerator}}{ ext{leading coefficient of denominator}}$

  • If degree numerator > degree denominator: No horizontal asymptote

• Oblique (slant) asymptotes: If degree numerator is exactly one more than denominator, divide numerator by denominator.

Solving Inequalities

Polynomial and Rational Inequalities

Solving inequalities involves finding intervals where the function is positive or negative.

• Set the inequality: $f(x) > 0$ or $f(x) < 0$

• Find zeros: Solve $f(x) = 0$

• Test intervals: Use test points in each interval determined by the zeros.

• Express solution: Use interval notation.

• Example: Solve $x^2 + 7x  2 > 0$

Summary Table: Asymptotes of Rational Functions

Type	How to Find	Example
Vertical	Set denominator = 0	$x = 1$ for $f(x) = \frac{1}{x-1}$
Horizontal	Compare degrees	$y = 0$ for $f(x) = \frac{1}{x^2}$
Oblique	Divide numerator by denominator (if degree numerator = degree denominator + 1)	$y = x$ for $f(x) = \frac{x^2}{x}$

Additional info:

• Some questions involve graph interpretation and construction, which are essential skills in precalculus for understanding function behavior.

• Interval notation is used throughout to express domains and solution sets.

• Complex zeros and factoring are included, which are relevant for higher-level polynomial analysis.