Polynomial Functions Cheat Sheet

Master the language, manipulation, and graphical representation of polynomial functions, from basic definitions to solving complex equations and inequalities.

Core Principles

Polynomials are expressions with variables and coefficients, involving only non-negative integer exponents.
The degree of a polynomial determines its general shape and behavior.
Polynomial division is a method to divide a polynomial by another polynomial of a lower degree.
Factoring polynomials simplifies expressions and aids in finding roots.
Cubic equations (degree 3) can have up to three real roots.
Transformations (translations, dilations, reflections) alter the position and scale of function graphs.

Action Steps

Identify the degree of the polynomial.
Use polynomial long division or synthetic division to divide polynomials.
Apply factoring techniques (grouping, difference of squares, sum/difference of cubes) to factor polynomials.
Use the Rational Root Theorem and synthetic division to find roots of cubic and higher-degree polynomials.
Graph factorized cubic functions by identifying roots and end behavior.
Solve cubic inequalities by finding roots and testing intervals.
Apply transformations to sketch or analyze function graphs.

Formulas

Polynomial form: $a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$
Cubic function form: $f(x) = a(x-h)^3 + k$
Translation: $f(x-h) + k$ shifts the graph h units horizontally and k units vertically.
Dilation (y-axis): $a f(x)$ stretches or compresses the graph vertically by a factor of |a|.
Dilation (x-axis): $f(x/b)$ stretches or compresses the graph horizontally by a factor of |b|.
Reflection (x-axis): $-f(x)$ reflects the graph across the x-axis.
Reflection (y-axis): $f(-x)$ reflects the graph across the y-axis.

Key Terms

Polynomial: An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
Degree: The highest exponent of the variable in a polynomial.
Root (or Zero): A value of x for which a polynomial equals zero.
Factor: An expression that divides another expression evenly.
Cubic Function: A polynomial function of degree three.
Quartic Function: A polynomial function of degree four.
Transformation: Operations that move, resize, or reflect a graph (translation, dilation, reflection).

Real World Examples

Modeling projectile motion: Quadratic and cubic polynomials are used to describe the path of objects thrown or launched.
Economic forecasting: Polynomial functions can model trends in data for predictions.
Engineering design: Polynomials help in designing curves for bridges, roads, and other structures.

Timeline

Ancient Greece: Early work on solving polynomial equations, particularly cubic equations (e.g., Archimedes).
16th Century: Development of general formulas for solving cubic and quartic equations (Cardano, Ferrari).
17th Century: René Descartes links algebra and geometry, introducing polynomial notation and graphing.
19th Century: Évariste Galois develops group theory, providing insights into the solvability of polynomial equations.
Modern Era: Continued application of polynomial functions in calculus, computer graphics, and data analysis.

People

René Descartes: Pioneered the use of algebraic notation and graphing of polynomials.
Geronimo Cardano: Published methods for solving cubic equations.
Lodovico Ferrari: Developed a method for solving quartic equations.
Évariste Galois: Developed theory related to the solvability of polynomial equations.