High School Math-Advanced Function-Polynomial Function  

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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