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

# Zeros of polynomials & their graphs

## Video transcript

use the real zeros of the polynomial function y is equal to X to the third plus 3x squared plus X plus three to determine which of the following could be its graph so there's a several ways of trying to approach it one we could just look at where what the zeros of these graphs are what they appear to be and then see if this function is actually zero when X is equal to that so for example in graph a and first of all I always I encourage you to pause this video and try it before I about to I before I go out before I show you how to solve it so I'm assuming you've given a go at it so let's look at this first graph here it's zero it clearly has a zero at right at this point and just by trying to inspect to this graph it looks like this is that X is equal to negative three if I were to estimate so that looks like the point negative three zero so let's see if if we substitute x equals negative three here whether we get Y equaling zero so let's see why let's see negative three to the third power plus three times negative three squared plus negative three plus three what does this give us this gives us negative twenty-seven this gives us positive 27 this gives of course negative three this is plus three these two cancel out these two cancel out this does indeed equal zero so this was actually pretty straightforward graph a does indeed work you could try graph B right here and you would have to verify that we have a zero at negative this looks like negative two another one this looks like at one another one that looks at three and it'll end if and since we already know that a is the answer none of these if you really input x equals negative two x equals 1 or x equals three into this function definition right over here you should not get zero and you'll see that this doesn't work same thing for this one if you tried four or seven for your X's you should not get zero over here because we see that the real function does not equal zero at four or seven another giveaway that this is not going to be the function is that you're going to have a total you're going have a total of three routes let me write this down so you're going to have a total of three routes three routes now those three routes could be real or complex roots so and the the big key is complex roots come in pairs complex roots in pairs so you might have a situation situation with three real roots and this is an example with three real roots although we know this actually isn't the function right over here or if you have one complex root you're going to have another complex root so if you have if you have any complex roots the next possibility is one real and two complex roots two complex roots and this right over here has two real roots that's not a possibility that would somehow imply that you have only one complex you only have one complex root which is not going to be a it's not going to that that is not a possibility now another way that you could have thought about this and this would have been the longer way but let's say you didn't have the graphs here for you and someone asks you to just find the roots well you could have attempted to factor this and this one actually is factorable Y is equal to X to the third plus 3x squared plus X plus 3 as mentioned in previous videos factoring things of degree higher than two there is something of an art to it but oftentimes if someone expects you to you might be able to group things in interesting ways especially when you see that several terms have some common factors so for example these first two terms right over here these first two terms right over here have the common factor x squared so if you were to factor that out you get x squared times X plus 3 which is neat because that looks a lot like the second two terms we could write that as plus 1 times X plus 3 and then you can factor the X plus 3 out we could factor the X plus 3 out and we would get X plus 3 times x squared times x squared plus 1 x squared plus 1 and now your zeroes are going to happen or this whole wide remember this is equal to Y y is going to be equal if Y is going to equal zero if either one of these factors is equal to zero so when does X plus 3 equals zero well subtract 3 from both sides that happens when X is equal to negative 3 when does x squared equal 0 when does x squared is x squared plus 1 equals 0 I should say well when x squared is equal to negative 1 well there's no real X's no real valued X's there's no real number X is such that x squared is equal to negative 1 X is going to be X is going to be an imaginary or I guess I'll just save more general terms it's going to be complex valued so once again you see this you're going to have a pair of complex roots and you have one you have one real root at X is equal to negative 3