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  • - So we're given a p of x, it's a third degree polynomial,

  • and they say, plot all the zeroes or the x-intercepts

  • of the polynomial in the interactive graph.

  • And the reason why they say interactive graph,

  • this is a screen shot from the exercise on Kahn Academy,

  • where you could click and place the zeroes.

  • But the key here is, lets figure out what x values make

  • p of x equal to zero,

  • those are the zeroes.

  • And then we can plot them.

  • So pause this video,

  • and see if you can figure that out.

  • So the key here is to try to factor this expression

  • right over here, this third degree expression,

  • because really we're trying to solve the X's

  • for which five x to third plus five x squared

  • minus 30 x is equal to zero.

  • And the way we do that is

  • by factoring this left-hand expression.

  • So the first thing I always look for

  • is a common factor across all of the terms.

  • It looks like all of the terms are divisible by five x.

  • So let's factor out a five x.

  • So this is going to be five x times,

  • if we take a five x out of five x to the third,

  • we're left with an x squared.

  • If we take out a five x out of five x squared,

  • we're left with an x, so plus x.

  • And if we take out a five x of negative 30 x,

  • we're left with a negative six is equal to zero.

  • And now, we have five x times this second degree,

  • the second degree expression and to factor that,

  • let's see, what two numbers add up to one?

  • You could use as a one x here.

  • And their product is equal to negative six.

  • And let's see, positive three and negative two

  • would do the trick.

  • So I can rewrite this as five x times,

  • so x plus three, x plus three,

  • times x minus two, and if what I did looks unfamiliar,

  • I encourage you to review factoring quadratics

  • on Kahn Academy,

  • and that is all going to be equal to zero.

  • And so if I try to figure out what x values

  • are going to make this whole expression zero,

  • it could be the x values

  • or the x value that makes five x equal zero.

  • Because if five x zero,

  • zero times anything else is going to be zero.

  • So what makes five x equal zero?

  • Well if we divide five, if you divide both sides by five,

  • you're going to get x is equal to zero.

  • And it is the case.

  • If x equals zero, this becomes zero,

  • and then doesn't matter what these are,

  • zero times anything is zero.

  • The other possible x value that would make everything zero

  • is the x value that makes x plus three equal to zero.

  • Subtract three from both sides

  • you get x is equal to negative three.

  • And then the other x value is the x value that makes

  • x minus two equal to zero.

  • Add two to both sides, that's gonna be x equals two.

  • So there you have it.

  • We have identified three x values that make our polynomial

  • equal to zero and those are going to be the zeros

  • and the x intercepts.

  • So we have one at x equals zero.

  • We have one at x equals negative three.

  • We have one at x equals, at x equals two.

  • And the reason why it's,

  • we're done now with this exercise,

  • if you're doing this on Kahn Academy

  • or just clicked in these three places,

  • but the reason why folks find this to be useful is

  • it helps us start to think about what the graph could be.

  • Because the graph has to intercept

  • the x axis at these points.

  • So the graph might look something like that,

  • it might look something like that.

  • And to figure out what it actually does look like

  • we'd probably want to try out a few more x values

  • in between these x intercepts

  • to get the general sense of the graph.

- So we're given a p of x, it's a third degree polynomial,

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