![]() ![]() Has a multiple of one, only one of the expressions There are only, they only deduced one time when you look at it in factored form, only one of the factors And so for each of these zeros, we have a multiplicity of one. ![]() And this notion of having multiple parts of our factored form that wouldĪll point to the same zero, that is the idea of multiplicity. In some ways you could say that hey, it's trying to reinforce that we have a zero at x minus three. We actually have two zeros for a third degree polynomial, so something very Zero at x equals three," but we already said that, so Of the expression would say, "Oh, whoa we have a To be equal to zero, and we can see that it intersects the x axis at x equals three. And then if x is equal to three, this whole thing's going White graph also intersects the x axis at x equals one. Going to be equal to zero, so we have zero at x equals one, and we can see that our So our zeros, well once again if x equals one, this whole expression's Zeros does P2 have? Pause this video and think about that. But how many zeros, how many distinct unique Have an x to the third term, you would have a third degree polynomial. Now what about P2? Well P2 is interesting, 'cause if you were to multiply this out, it would have the same degree as P1. So between these first two, or actually before thisįirst zero it's negative, then between theseįirst two it's positive, then the next two it's negative, and then after that it is positive. Zeros our function, our polynomial maintains the same sign. We can also see the property that between consecutive And we can see it here on the graph, when x equals one, the graph of y is equal to ![]() When x is equal to two,īy the same argument, and when x is equal to three. To be equal to zero because zero times anything is zero. When x is equal to one, the whole thing's going Polynomial is equal to zero and that's pretty easy toįigure out from factored form. Just make it the zeros, the x values at which our What we're going to do in this video is continue our study of zeros, but we're gonna look at a special case when something interesting This is the graph of Y isĮqual to P1 of x in blue, and the graph of Y isĮqual to P2 x in white. And they have beenĮxpressed in factored form and you can also see their graphs. So what we have here are two different polynomials, P1 and P2. When we drive across x = -2, the factors become (positive)(negative)(negative), so the polynomial becomes positive.īut when we drive across x = 0, BOTH of the remaining factors flip at the same instant, so the factors become (positive)(positive)(positive), and the polynomial stays positive. When we drive in from the left, the three factors start out as (negative)(negative)(negative), so the polynomial is negative. So in that case, the sign of the polynomial DOESN'T change.įor example, say we have x^3 + 2x^2 = (x + 2)(x)(x). The very instant you cross x = 2, the polynomial becomes (positive)(negative)(negative) = positive.Įvery time you "drive across" a zero, exactly one of the linear factors changes sign from negative to positive, and that flips the sign of the polynomial.īut when you have two identical roots, then TWO of the factors change sign from negative to positive at the same instant. The first linear factor, (x + 2), goes from negative to zero to positive. When you multiply three negative numbers together, you get a negative result, so the entire polynomial will come out negative. As you drive onto the screen from the left, all three factors will be negative numbers.įor example, if x = -100, the polynomial will equal (-100 + 2)(-100)(-100 - 2) = (negative)(negative)(negative). ![]() Imagine you are driving along the number line from left to right. You can break a polynomial into "linear factors." For example, we can break x^3 - 4x into (x + 2)(x)(x - 2). ![]()
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