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  • - [Instructor] So we're told that p of x

  • is equal to this expression here,

  • and it says plot all the zeros or x-intercepts,

  • of the polynomial in the interactive graph,

  • and the reason why it says interactive graph

  • is this is a screenshot from this type

  • of exercise on Khan Academy,

  • and on Khan Academy, you'd be able to click on points here

  • and it'd put little dots

  • and you can either delete them and put 'em someplace else,

  • so it'd be an interactive graph.

  • But this is just a screenshot,

  • so I'm just going to draw on top of this.

  • But the main goal is, what are the zeros of this polynomial

  • and then you just to plot it on this graph,

  • so pause this video and have a go at it.

  • All right, now to figure out the zeros of a polynomial,

  • you would essentially have to figure out the x values

  • that would make the polynomial equal to zero.

  • Or another way to think about it is the x values

  • that would make this equation true.

  • X to the third plus x squared

  • minus nine x minus nine is equal to zero.

  • Now, the best way to do that

  • is to try to factor this expression.

  • Now, this is a third degree polynomial,

  • which isn't always so easy to factor,

  • so let's see how we might approach it.

  • The first thing I look for is are there any common factors

  • to all of these terms, and it doesn't look like there is.

  • The next thing I could look for,

  • I could think about whether factoring

  • by grouping could work here,

  • and when I think about factoring by grouping,

  • I would look at the first two terms and I would

  • look at the last two terms,

  • and I would say, is there anything I could factor out

  • of these first two terms that would,

  • or what's the most that I could factor out

  • of these first two terms,

  • and then what's the most that I could factor out

  • of these last two terms,

  • and then it would leave

  • something similar once I've done that factoring.

  • Now, what I mean is, for these first two,

  • we have a common factor of x squared,

  • so let's factor out an x squared

  • and these first two terms become x squared times x plus one,

  • and then for these second two terms,

  • I can factor out a negative nine,

  • so I could rewrite it as negative nine times x plus one.

  • Now, that all worked out quite nicely,

  • because now we see, if we view,

  • if we view this as our now our first term

  • and this as our second term,

  • we can see that x plus one is a factor of both of them.

  • And so we can factor that out.

  • We can factor out the x plus one,

  • and I'll do that in this light blue color,

  • actually let me do it with slightly darker blue color.

  • And so if you factor out the x plus one,

  • you're left with x plus one times x squared,

  • x squared, minus nine.

  • Minus nine.

  • And that is going to be equal to zero.

  • Now, we are not done factoring yet,

  • because now we have a difference of squares.

  • X squared minus nine, this is going to be equal to,

  • and let me just write it all out,

  • so I have this x plus one here,

  • so I have x plus one, and then the x squared minus nine,

  • I can write as x plus three times x minus three.

  • If any of what I'm doing feels unfamiliar to you,

  • if that first factoring feels unfamiliar,

  • I encourage you to review factoring by grouping,

  • and if what I just did looks unfamiliar,

  • I encourage you to look at factoring differences of squares.

  • But anyway, all of that would be equal to zero.

  • Now, if I have the product of several things equaling zero,

  • any, if any one of those things is equal to zero,

  • that would make the whole expression equal to zero.

  • So we have a situation where one

  • solution would be the solution

  • that makes x plus one equal to zero,

  • and once again, I'm gonna do that in darker color,

  • x plus one equal zero, and

  • that of course is x is equal to negative one.

  • Another solution is what would make x

  • plus three equal to zero?

  • And that of course is x is equal to negative three,

  • subtract three from both sides,

  • and then another solution is

  • going to be whatever x value makes x minus three equal

  • to zero, add three to both sides,

  • you get x is equal to three.

  • So there you have it.

  • We have our three zeros.

  • Our polynomial evaluated

  • at any of these x values will be equal to zero.

  • So we can plot it here on this interactive graph,

  • I'm just gonna draw on it.

  • So we have x equals negative one,

  • which is right over there,

  • x equals negative three,

  • which is right over there,

  • and x equals three, which is right over there.

  • And the reason why you might want to do this type of thing,

  • this exercise just asks us to do this,

  • and we're done, but the reason why this

  • is useful is this can help inform what the graph looks like.

  • This tells us where our graph intersects the x-axis.

  • So our graph might do something like this,

  • or it might do something like this,

  • and we would have to look

  • at other information to think about what that might be.

  • But I'll leave you there.

- [Instructor] So we're told that p of x

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