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.

# Zeros of polynomials: plotting zeros

When we are given a polynomial in factored form, we can quickly find the polynomial's zeros. Then, we can represent them as the x-intercepts of the polynomial's graph.

## Want to join the conversation?

• I'm from Russia and berly know English so if you can help me understand the first part.....Thanks... If you can translate it to English that would be great.. I use google translate.
• So basically every polynomial function has "zeros" and these are also called x-intercepts. Zeros are when a polynomial function "intersects" or touches the x-axis. When a polynomial is in factored form, like the question in the video, it is very easy to find the zeros. If you think about it, an x-intercept is when a function intersects the x-axis, and for this to be true, the y-value of that coordinate must be equal to zero. So to solve we can use this property -
If (A)(B)(C) = 0
Then either A, B or C must be = 0
So in the case of 2x(2x+3)(x-2), we just set "A" "B" and "C" as equal to zero, when -
A = 2x = 0
B = 2x+3 = 0
C = x-2 = 0
Now we just solve for x to get our zeros!
We are now left with
x = 0
x = -3/2
x = 2
I hope this helped, If this confused you more or if something seems unclear, please let me know, I'm happy to help!
• why do we need to equal the polynomial to 0 always? why cant it be some other number?
• They're all equivalent. For example, finding out when x³-x+1=5 is the same problem as finding out when x³-x-4=0.

We choose 0 to standardize on because if a product of several things is 0, then one of the factors must itself be 0. No other number has this property. This means that factoring the polynomial and finding its solutions are now the same problem.
• Would it be valid to say that instead of the zeroes of the polynomial being the x values that make the polynomial equal to zero, that they could be the x-intercepts that make y = 0 because the polynomial could be written as y = p(x) right?
(1 vote)
• Yes, they would also be seen as the x-intercepts, technically. But you wouldn't need to say that they are the x-intercepts that "make y=0" because that is the definition of an x-intercept.

## Video transcript

- [Instructor] We're told we want to find the zeros of this polynomial and they give us the polynomial right over here, and it's in factored form. And they say plot all the zeros, or the x-intercepts, of the polynomial in the interactive graph. And so this is a screenshot from Khan Academy. If you're doing it on Khan Academy, you would click where the zeros are to plot the zeros, but I'm just gonna draw it in. So pause this video and see if you could have a go at this before we work on this together. All right, now let's work on this together. So the zeros are the x values that make our polynomial equal to zero. So another way to think about it is for what x values are p of x equal to zero? Those would be the zeros. So essentially, we have to say, hey, what x values would make two x times two x plus three times x minus two, 'cause this is p of x, what x values would make this equal to zero? Well, as we've talked about in previous videos, if you take the product of things and that equals zero, if any one of those things equal zero, at least one of those things equal zero, make the whole product equal zero. So for example, if two x is equal to zero, it would make the whole thing zero, so two x could be equal to zero, and if two x is equal to zero, that means x is equal to zero, and you could try that out. If x is equal to zero, this part right over here is going to be equal zero. Doesn't matter what these other two things are. Zero times something times something is going to be equal to zero. And then you could say, well, well maybe two x plus three is equal to zero, so we could just write that. Two x plus three is equal to zero, and if that were true, what would x, or what would x have to be in order to make that true? Subtract three from both sides, two x would have to be equal to negative three, or x would be equal to negative 3/2. So this is another x value that would make the whole thing zero, 'cause if x is equal to negative 3/2, then two x plus three is equal to zero, you take a zero times whatever this is and whatever that is, you're gonna get zero. And then last but not least, x minus two could be equal to zero. That would make the whole product equal to zero. So what x value makes x minus two equal zero? We'll add two to both sides, and you would get x is equal to two. If x equals two, that equals zero, doesn't matter what these other two things are. Zero times something times something is going to be equal to zero. So just like that, we have the zeros of our polynomial, and the reason why they have x-intercepts in parentheses here is that's where the graph of p of x, if you say y equals p of x, that's where it would intersect the x-axis, and that's because that's where our polynomial is equal to zero. So let's see, we have x equal zero which is right over there. Once again, if you're doing this on Khan Academy, you would just click right over there and it would put a little dot there. We have x is equal to negative 3/2, which is the same thing as negative 1/2, so that's right over there. And then, we have x equals two, which is right over there. So those are the x-intercepts or the zeros of that polynomial. Now, this is useful in life, because you could use it to graph a function. I don't know exactly what this function looks like, maybe it looks something like this, maybe it looks something like this. We would have to try out a few other values to get a sense of that, but we at least know where it's intersecting the x-axis. It's at the zeros.