Engineering Math - Matrix

What is Polynomial Function ?

Polynomial Function is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s). it can be express in a generic form as shown below.

How to Graph ?

You can graph any polynomial functions by both hands(harder method) and technology(easier method).

When graphing with hands, you need to know the x-intercepts(also known as roots), y-intercept, and end behaviours of the function. Some of the tips you can use for drawing by hands are

• When a polynomial function is in its standard form, you must factor the equation in order to find the roots
• You can use both synthetic division and long division to factor the function
• If a polynomial function is an even degree function with positive sign, the end behaviour is quadrant 2 to quadrant 1. However if the function has a negative sign, the end behaviour is quadrant 3 to quadrant 4
• If a polynomial function is an odd degree function with positive sign, the end behaviour is quadrant 3 to quadrant 1. However if the function has a negative sign, the end behaviour is quadrant 2 to quadrant 4
• End behaviours are ALWAYS from left to right

< Quadrant on a Cartesian Coordinate >

Descartes rule of signs

This rule is used to identify the number of positive roots, negative roots, and/or imaginary Roots of a polynomial function. By the application of this rule, you can identify whether you found the correct roots or not.  

• If a polynomial function of f(x) has n sign changes, there are n positive roots
• If f(-x) has a sign changes, there are a negative roots
• Degree of function = total Number of positive and Negative root(s) and number of imaginary root(s)

Even and odd functions

Odd function is an odd-degree function with a point symmetry, proved by f(-x)=-f(x), x can be any real numbers

Even function is an even-degree function with a line symmetry, proved by f(x) = f(-x), x can be any real numbers

Polynomial Division

< Synthetic Division >

Write the answer in P(x)Q(x)+R(x) format