Graph each polynomial function. Factor first if the polynomial is...

Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=x^2(x-5)(x+3)(x-1)

Accepted Answer

Hello. Today we're gonna be graphing the polynomial function. So the polynomial function given to us is F of X is equal to X squared times the quantity X minus two times the quantity X plus two times the quantity X minus seven. Now, the first thing we're gonna want to do is to figure out the X intercepts for the polynomial function. In order to do that, we're gonna go ahead and set the function equal to zero. And if we use our zero theorem, we can set each of the factors of X equal to zero in order to solve before the X intercepts. So for the X intercepts, we're gonna end up with four equations. We're going to get X squared is equal to zero, X minus two is equal to zero, X plus two is equal to zero and X minus seven is equal to zero. If we take the square root of both sides of the first equation, we get X is equal to zero. If we add two to both sides of this equation, we get X is equal to two. If we subtract two to both sides of the third equation, we get X is equal to negative two. And if we add seven to both sides of the last equation, we get X is equal to seven. So in total, we have four X intercepts, we have X is equal to 02, negative two and seven. Now, what we want to do is to determine the white intercept. Now, since the Y intercept lies on the Y axis, this is going to be the region where X is equal to zero. So the Y intercept is going to exist where every value of X is equal to zero. But notice that if we set the first value of X equal to zero, the entirety of the polynomial function is going to be zero. What this means is that the Y intercept is going to equal to zero and it's going to be located at the origin zero comma zero. Now, what we want to do is we want to find the regions of increasing and decreasing for the polynomial function. In order to do that, we're gonna first draw a number line going from negative infinity to positive infinity. And what we're going to label on this number line is all the values of the X intercepts. So we're going to have negative 20, two and seven. And now what we wanna do is we want to take a testing point from each of these regions to plug into our function and use that to determine the regions of increasing and decreasing. Now, there is a shortcut that we can do to find the behaviors of the end points. See if we were to multiply all these values together, we're going to get a polynomial whose leading coefficient is going to be X to the fifth. This is going to be a positive leading coefficient with an odd powered exponent. And since the exponent is odd, that means that this polynomial function is going to be odd. That means that the region as it approaches negative infinity is going to decrease and the region as it approaches positive infinity is going to increase. So with that tells us the end behaviors as seven approaches infinity which is positive and as negative two approaches negative infinity which is negative. And so what we really need to do now is to take the testing points of the three regions between negative two and 00 and two and two and seven. So let's take a testing point between negative two and zero. And that testing point can be negative one. When we plug in negative one into our function, we get the output of 24 since this is positive, that tells us that this region is going to increase. Now let's take a testing point between zero and two. And that testing point can be when F is equal to one. So if we plug in one into our function, we get the output value of 18 since this is positive that means the region between zero and two is also going to be increasing. And finally, we need a testing point between two and seven. And that testing point can be three when we plug in three into our function, that is going to give us an output of negative 180. And that means that this region is going to be decreasing. So now that we have the X intercepts Y intercept and the regions of increasing and decreasing. Let's go ahead and plot all the information together. So we're going to draw an axis and we're going to plot all of our points. We're going to have the X intercepts at negative two at the origin zero at positive two and at positive seven. And we can use these as a guideline to figure out what the rough shape of the graph is going to be. See, we already know that the region after seven is going to be increasing to infinity. So that means that the end behavior is gonna roughly look something like this and we know that the region as it decreases past negative two is going to approach negative infinity. So we're gonna have a rough shape like this for the end behaviors. Now we know that the graph is going to be positive between the region of negative two and zero. So we can represent that with a positive hill. And we also know that the region is going to be positive going from 0 to 2. So that means that the region is going to bounce off a zero and connect to two. And finally, the region is going to be negative or decreasing between two and seven. So it's gonna dip below the X axis then reconnect with seven afterward. And this is going to be the rough shape of the polynomial function. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=x^2(x-5)(x+3)(x-1)

Key Concepts

Polynomial Functions

Factoring Polynomials

Graphing Polynomial Functions

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