Polynomial and Rational Functions

Quadratic Functions & Models

Quadratic functions are fundamental in precalculus, modeling various real-world phenomena such as projectile motion. The general form of a quadratic function is , where . The graph of a quadratic function is a parabola, which is symmetrical about a vertical line called the axis of symmetry. The vertex is the intersection of the parabola and its axis of symmetry.

• Definition: A quadratic function is any function of the form .

• Graph: The graph is a parabola. If , the parabola opens upward and the vertex is a minimum. If , it opens downward and the vertex is a maximum.

• Axis of Symmetry: The axis of symmetry is .

• Vertex: The vertex is at .

• Standard Form: , where is the vertex.

• x-Intercepts: Found by solving .

• Example: has vertex at .

Polynomial Functions of Higher Degree

Polynomial functions generalize quadratics to higher degrees. The degree of a polynomial is the highest power of with a nonzero coefficient. The leading coefficient is the coefficient of the highest degree term.

• Definition: , where .

• Degree: The degree is .

• Coefficients: is the leading coefficient, is the constant term.

• Example: is degree 5; is degree 0.

• Not a Polynomial: If a variable is raised to a non-integer or negative power, it is not a polynomial.

Polynomials and Synthetic Division

Polynomial division is used to simplify expressions and find zeros. Long division is similar to numerical division, while synthetic division is a shortcut for divisors of the form .

• Long Division:

• Synthetic Division: Efficient for divisors . Steps: list coefficients, use , perform operations to get quotient and remainder.

• Division Algorithm: , where has degree less than .

• Example: Divide by using synthetic division.

The Remainder Theorem and Factor Theorem

These theorems connect division and zeros of polynomials.

• Remainder Theorem: If is divided by , the remainder is .

• Factor Theorem: is a factor of if and only if .

• Application: Use synthetic division to test if is a factor.

Complex Numbers

Complex numbers extend real numbers to include solutions to equations like . The imaginary unit is defined as .

• Standard Form: , where is real, is imaginary.

• Equality: if and only if and .

• Operations:

  • Addition:

  • Subtraction:

  • Multiplication:

  • Complex Conjugate: and ;

• Quadratic Equations: Use the quadratic formula ; if , solutions are complex.

Zeros of a Polynomial Function

Zeros of a polynomial are values of where . These can be real, rational, irrational, or complex.

• Rational Zero Test: Possible rational zeros are , where is a factor of the constant term and is a factor of the leading coefficient.

• Conjugate Pairs: If is a zero, is also a zero for polynomials with real coefficients.

• Linear Factorization Theorem: Any polynomial of degree can be written as , where are complex numbers.

Non-linear Inequalities

Polynomial and rational inequalities involve finding intervals where the function is positive or negative.

• Polynomial Inequality: Find zeros, divide the number line into intervals, test each interval, and construct a sign chart.

• Rational Inequality: Consider zeros and undefined values (where denominator is zero).

• Domain: The set of all -values for which the expression is defined.

• Example: Solve by finding zeros and testing intervals.

Summary Table: Polynomial Function Properties

Additional info:

• Some examples and step-by-step procedures were expanded for clarity.

• Images were included only when directly relevant to the mathematical explanation.