End Behavior of Polynomial Functions End Behavior of Polynomial Functions means to find what will be the characteristics of the function at extreme points i.e. at points when x approaches to on positive as well as negative axis. End behavior can be found with the help of graphs. It can be directly manipulated by seeing the graph that which part of the graph will go up and which one will go down at the extreme points. End behavior can also be found by applying some rules on the function equations. Both methods of finding the end behavior are described with the help of proper examples in the following section. Finding End Behavior of Polynomials Consider a polynomial f (x) = anxn + an-1xn-1 + ... + a2x2 + a1x + a0 Know More About Factoring Trinomial
Now we can find the end behavior by just knowing the values of Leading Co-efficient an and Power n of the Polynomial Equation. Feasible Cases :Case 1:- If Leading coefficient(an) is positive, and the power(n) is even, both ends of the graph go up. Case 2:- If Leading coefficient(an) is negative, and the power(n) is even, both ends of the graph go down Case 3:- If Leading coefficient(an) is positive, and the power(n) is odd, the right hand of the graph goes up and the left hand goes down Case 4:- If Leading coefficient(an) is negative, and the power(n) is odd, the right hand of the graph goes down and the left hand goes up. Examples for Graphical End Behavior Below are some examples for graphical end behavior Example 1: Let us take an graphical example to show end behavior :Learn More About Operations with Polynomials
It is clear from the graph that that when x approaches to , the right part of the graph is going upward. Similarly, when x approaches to -then left part of the graph is also going upward. Example 2: This example is somewhat different from the previous one. Here, when x approaches to , the right part of graph extends towards upward direction. But, when x approaches to - , the left part of the graph extends towards downward direction. Example 3: f(x)= - 3x9+7 Here, Leading Co-efficient = -3 = -ve value Degree of polynomial= 9 = ODD Using CASE 4 described above, we can conclude that the right hand of the graph goes down and the left hand goes up. End Behavior:- The right part of the graph goes down and the left hand goes up.
How to Simplify Polynomials How to Simplify Polynomials? The form of an monomial is expression is a(xn) where n is non-negative integer. The variable ‘a’ is called as coefficient of xn and n is the degree of the monomial. Based on the n value monomial is called as monomial (when n=1), two degree polynomial (when n=2) and three degree polynomial (when n=3). Example: Monomial = x2 Binomial =3x2+2x Trinomial =5x4+3x2+8 Zero polynomial = all the coefficient of polynomial is zero called as zero polynomial We can do the following operations in internet solving polynomials
Addition of polynomials Subtraction of polynomials Multiplication of polynomials Types of Polynomials Linear polynomials Quadratic polynomials Cubic polynomials Bi-quadratic polynomials Internet solving polynomials: The following sums are the example for internet solving polynomial Polynomial Examples Below are some examples on polynomials Example 1: Add the following two polynomials 5x3+3x2+2x+1 and 6x2+3x+2 Solution: Given: 5x3+3x2+2x+1 and 6x2+3x+2
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