Determine the end behavior of the given polynomial function. f ( x ) = x 2 + 4 x + x + 7 x 3 f\left(x\right)=x^2+4x+x+7x^3 f ( x ) = x 2 + 4 x + x + 7 x 3

In this practice problem we want to determine that the end behavior of the given polynomial function, and here I have the function f of x is equal to x squared plus 4x plus x plus 7x cubed. Now remember, to determine end behavior, we want to look at the very first term when our function is written in standard form. But looking at the polynomial function that I have here, it's actually not written in standard form. So that is the very first thing I wanna do is rewrite this in standard form. Now, remember, we wanna have all like terms combined. So here I have this 4 x and this x that can get combined, and then I need it to be in descending order of power, which I can do by moving this 7x cube to the front. So let's go ahead and rewrite this in standard form. I end up with f of x is equal to 7x cubed plus x squared. And then lastly, combining that 4x and that x give me positive 5x. Now double checking here, I am in standard form now. I'm in descending order of power from 3 to 2 to a power of 1, and everything that can be combined is combined, so we're good. Now we can go ahead and determine our end behavior by looking at that very first term in standard form, which here is 7x3. Now, first, to determine that the behavior of the right side of our function, I know that I need to look at my leading coefficient, which here is 7. Now, this is a positive 7. If I have a positive leading coefficient, I know that that right side is going to rise up, so my right side rises. Now to determine the behavior of the other end, so the ends, remember, I'm looking at the degree here, which which in this case is 3. Now 3 is an odd number, and remember odd opposite. So my n's are going to have the opposite behavior. That's all for this one. Thanks for watching.

4. Polynomial Functions

Understanding Polynomial Functions

College Algebra4. Polynomial FunctionsUnderstanding Polynomial Functions

Multiple Choice

Determine the end behavior of the given polynomial function. $f\left(x\right)=x^2+4x+x+7x^3$

Right side rises; Ends are same

Right side rises; Ends are opposite

Right side falls; Ends are same

Right side falls; Ends are opposite

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Determine the end behavior of the given polynomial function. ﻿f(x)=x2+4x+x+7x3f\left(x\right)=x^2+4x+x+7x^3f(x)=x2+4x+x+7x3﻿

Determine the end behavior of the given polynomial function. $f\left(x\right)=x^2+4x+x+7x^3$