If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:9:09

CCSS.Math: , , ,

so we have a fifth degree polynomial here P of X and we're asked to do several things first find the real roots and let's remind ourselves what roots are so roots is the same thing as a zero and they're the X values that make the polynomial equal to zero so the real roots are the X values where P of X is equal to zero so the x values that satisfy this are going to be the roots or the zeros and we want the real ones as you'll learn in the future there's also going to be imaginary roots or zeros or there might be then we want to think about how many times how many times we intercept the x axis well as we'll see however many real roots we have that's how many times we are going to intercept how many however many unique real roots we have that's however many times we're going to intercept the x axis how do I know that well let's just think about an arbitrary polynomial here so those are my axes this is the x axis that's my Y axis and let me just graph an arbitrary polynomial here so let's say it looks like that well what's going on right over here at this x value we see based on the graph of the function that P of X is going to be equal to 0 so that's going to be a root this is also going to be a root because at this x value the function is equal to 0 at this x value the function is equal to 0 at this x value the function is equal to 0 if we're on the x axis then the Y value is 0 so the function is going to be equal to 0 this is the graph of y is equal Y is equal to P of X now to sorry this P of X but I'm just drawing some arbitrary P of X so there's some x value that makes the function equal to 0 well that's going to be a point at which we are intercepting the x axis so we want to know how many times we're intercepting the x axis as well see it's going to be the same number of real roots or the same number of real zeros we have and then they want us to figure out the smallest of those X intercepts and we'll figure it out for this for this particular for this particular polynomial so let me give myself a little bit more space so let's get to it so we really want to solve P of x is equal to 0 so we really want to set that right over there equal to zero and solve this so we want to solve this equation the X values that make this equal to zero if I input them into the function I'm going to get the function equaling zero all right so the first thing that might jump out at you is that all of these terms are divisible by X so I like to factor that out from the get-go so we can rewrite this as x times X to the X to the fourth power plus nine x squared minus two x squared minus 18 is equal to zero now there's something else that you might that might have jumped out of you'd actually just jumped out at me as I was writing this down is that we have to we had to third degree terms and after we factored out an X now we have to second degree terms now it might be tempting to just add these two together and actually that it would be a completely legitimate way of trying to factor this so that we can solve this equation but instead of instead of doing it that way we might take this as a clue that maybe they want us to then maybe we can factor by grouping remember factor by grouping you split up that middle degree term and see if you can reverse the distributive property twice so let's see if we can do that can we group together these first two terms and factor something interesting out and group together these second two terms and factor something interesting out and then maybe we can factor something out after that what am I talking about well this is going to be the same thing as x times well this one actually let me write a big parenthesis here this one right over here is the same thing as I can factor out an x squared so it's going to be x squared plus sign it's going to be x squared if i factor out an x squared I'm going to get an x squared plus 9 and then over here if i factor out a let's see negative two I don't want to if i factor out a negative two I'm going to get so minus two times I'm going to get an x squared plus nine again now this is interesting because this is telling us maybe we can factor out an x squared plus nine so let me factor out x squared plus nine from to these terms and I'm going to get I'm going to get X I'll leave these big green parentheses here for now if we factor out an x squared plus nine it's going to be x squared plus nine times x squared x squared minus two x squared minus two and I gave myself a little bit too much space so let me delete that so let me delete that right over there and then close the parentheses then close the parentheses and I can actually I can even get rid of those green parentheses now if I want to optimally make this a little bit simpler so so far we've been able to factor it as x times x squared plus nine times x squared minus doing the whole point that I'm factoring this is if I can find the product of a bunch of expressions equaling zero then I could say well the product of those expressions are going to be zero if one or more of those expressions are equal to zero and I can solve for X let's see this one's completely factored if we're this one is completely factored if we're thinking about real roots this one you can view it as a difference of squares if your view is if you view two as the square root of two squared so we can rewrite this as and of course all of this is equal to zero let me just write equals equals so we could write this is equal to x times times x squared plus 9 times let's see I can factor this business into X plus the square root of two times X minus the square root of two I'm just recognizing this as a difference of squares and once again we just want to solve this hole all of this business equaling zero all of this equaling zero so how can this equal zero well any one of these expressions if I take the product and if any one of them equals zero then I'm going to get zero so X could be equal to zero X could be equal to zero and that actually gives us a root when X is equal to zero this polynomial is equal to zero and that's pretty easy to verify let's seek an x squared plus nine equals zero x-squared plus nine equals zero well if you subtract 9 from both sides you get x squared is equal to negative 9 and that's why I said there's no real solution to this so no real let me write that no real solution there is there are some imaginary solutions but no real solutions now can X plus the square root of 2 equals zero X plus the square root of two equals zero sure if we subtract square root of two from both sides you get X is equal to the negative square root of two and can X minus the square root of 2 equals zero sure you add square root of two to both sides you get X is equal to the square root of two so there we have it we have figured out our zeros X could be equal to zero P of 0 is 0 P of negative square root of 2 is 0 and P of square root of 2 is equal to 0 so those are our zeros X is their zeros are at 0 negative squares of two and positive squares of 2 and so those are going to be the three times that we intercept the x axis and what is the smallest of those intercepts well the smallest number here is a negative square root negative square root of 2 and you could tackle it the other way you could you could you could take this part right over with which part this part right over here and you could add those two middle terms and then factor in a non grouping way and I encourage you to do that but just to see that this makes sense that the zeros really are the x-intercepts I've I went to Wolfram Alpha and I graphed this this polynomial and this is what I got so this is what I got right over here so whenever if you see a fifth degree polynomial say well it might it'll have as many as five real zeros but if it has some imaginary zeros it won't have five real zeros instead this one has three and that's because the imaginary zeros which we'll talk more about in the future they come in these conjugate pairs so if you don't have five real roots the next possibility is that you're going to have three real roots and if you don't have three real roots the next possibility is going to have one real root so that's an interesting thing to think about and so here you see your three your three real roots you see your three real roots which correspond to the X values at which the function is equal to zero which is where we have our x-intercepts