Polynomial and Rational Functions: Key Concepts and Graphs

Complex Numbers and Quadratic Equations

Complex Numbers

Complex numbers extend the real number system by introducing the imaginary unit i, defined as $i = \sqrt{-1}$. Any complex number can be written in the form a + bi, where a and b are real numbers.

• Standard Form: a + bi

• Operations: Addition, subtraction, multiplication, and division follow binomial rules.

• Conjugate: The conjugate of a + bi is a - bi. The product of a complex number and its conjugate is always a real number.

• Principal Square Root: For any positive real number b, $\sqrt{-b} = i\sqrt{b}$.

Quadratic Equations with Complex Solutions

Quadratic equations of the form $ax^2 + bx + c = 0$ can have complex solutions if the discriminant $b^2 - 4ac$ is negative. These solutions are always conjugate pairs.

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

• Discriminant: If $b^2 - 4ac < 0$, the equation has two complex conjugate solutions.

Quadratic Functions and Their Graphs

Standard Form and Vertex

A quadratic function can be written in standard form as $f(x) = a(x - h)^2 + k$, where (h, k) is the vertex. The graph is a parabola, symmetric about the line x = h.

• Direction: If a > 0, the parabola opens upward; if a < 0, it opens downward.

• Vertex: The vertex is at $(h, k)$ or $\left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right)$ for $f(x) = ax^2 + bx + c$.

• Intercepts: Find x-intercepts by solving $f(x) = 0$; y-intercept is $f(0)$.

• Axis of Symmetry: x = h or $x = \frac{-b}{2a}$.

• Domain: All real numbers; Range: $[k, \infty)$ if a > 0, $(-\infty, k]$ if a < 0.

Polynomial Functions and Their Graphs

Definition and Properties

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 coefficients are real numbers.

• Degree: The highest power of x; determines the function's behavior.

• Leading Coefficient: The coefficient of the highest power term.

• Graph Characteristics: Smooth and continuous, no sharp corners or breaks.

End Behavior: Leading Coefficient Test

The end behavior of a polynomial function depends on its degree and leading coefficient.

• Odd Degree: Positive leading coefficient: falls left, rises right. Negative leading coefficient: rises left, falls right.

• Even Degree: Positive leading coefficient: rises both ends. Negative leading coefficient: falls both ends.

Zeros, Multiplicity, and x-Intercepts

The zeros of a polynomial function are the values of x for which $f(x) = 0$. The multiplicity of a zero refers to how many times a factor appears in the factorization.

• Even Multiplicity: Graph touches x-axis and turns around.

• Odd Multiplicity: Graph crosses x-axis.

• Maximum Turning Points: n - 1, where n is the degree.

Properties of Roots of Polynomial Equations

• If a polynomial equation is of degree n, it has n roots (counting multiplicities).

• If $a + bi$ is a root and coefficients are real, $a - bi$ is also a root (conjugate pairs).

The Linear Factorization Theorem

Any nth-degree polynomial can be written as a product of n linear factors (possibly complex), multiplied by a constant.

• $f(x) = a_n(x - c_1)(x - c_2)\ldots(x - c_n)$

• Each linear factor corresponds to a root (zero) of the polynomial.

Descartes' Rule of Signs

Descartes' Rule of Signs helps estimate the number of positive and negative real zeros of a polynomial function by counting sign changes in the coefficients.

• Positive Real Zeros: Number of sign changes in $f(x)$ or less by an even integer.

• Negative Real Zeros: Number of sign changes in $f(-x)$ or less by an even integer.

Rational Functions and Their Graphs

Definition and Domain

A rational function is a quotient of two polynomial functions: $f(x) = \frac{p(x)}{q(x)}$, where $q(x) \neq 0$. The domain excludes values that make the denominator zero.

Vertical and Horizontal Asymptotes

• Vertical Asymptote: x = a, where $q(a) = 0$ and $p(a) \neq 0$.

• Horizontal Asymptote: Determined by degrees of numerator (n) and denominator (m): - If n < m: y = 0 - If n = m: y = (leading coefficient of numerator)/(leading coefficient of denominator) - If n > m: No horizontal asymptote

Basic Reciprocal Functions

• Odd Function: $f(x) = \frac{1}{x}$, origin symmetry.

• Even Function: $f(x) = \frac{1}{x^2}$, y-axis symmetry.

Arrow Notation

Arrow notation is used to describe the behavior of functions as x approaches certain values or infinity.

Symbol	Meaning
x \to a^+	x approaches a from the right.
x \to a^−	x approaches a from the left.
x \to \infty	x approaches infinity; x increases without bound.
x \to -\infty	x approaches negative infinity; x decreases without bound.

Solving Polynomial and Rational Inequalities

Definitions and Procedure

A polynomial or rational inequality is an inequality involving a polynomial or rational function, such as $f(x) < 0$, $f(x) > 0$, $f(x) \leq 0$, or $f(x) \geq 0$.

• Express the inequality in standard form.

• Solve $f(x) = 0$ to find boundary points.

• Divide the number line into intervals using boundary points.

• Test values in each interval to determine where the inequality holds.

• Write the solution set, including boundary points if the inequality is non-strict ($\leq$ or $\geq$).

Summary Table: Polynomial vs Not Polynomial Functions

Type	Example	Reason
Polynomial	$f(x) = -3x^5 + \sqrt{2}x^2 + 5$	All exponents are nonnegative integers.
Not Polynomial	$F(x) = -3\sqrt{x} + \sqrt{2}x^2 + 5$	Exponent is not an integer.
Not Polynomial	$G(x) = -\frac{3}{x^2} + \sqrt{2}x^2 + 5$	Exponent is not a nonnegative integer.

Summary Table: Arrow Notation

Symbol	Meaning
x \to a^+	x approaches a from the right.
x \to a^−	x approaches a from the left.
x \to \infty	x approaches infinity; x increases without bound.
x \to -\infty	x approaches negative infinity; x decreases without bound.