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University of Toronto

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Problem 75

True or False? In Exercises $73-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $p(x)$ is a polynomial function, then $p$ has exactly one antiderivative whose graph contains the origin.

Answer

True, explanation inside.

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## Discussion

## Video Transcript

supposedly given upon a real P So p of X is a polynomial. And the question is, is there exactly one anti derivative anti which crosses the organs crossing origin? So it turns out that this is true on. We'll see why now. So it supposedly have a polynomial, and we take the anti derivatives. So just you notice like this particularly. Thank you. What we get is a different polynomial. Uh, and this polynomial has no constant term because what we've done is we've raised powers of X to take me on a derivative, and then we divide by than your parents. Uh, so we have this new polynomial, uh, plus C. And the idea here is that we can shoot, see in a specific way such that this anti derivative crosses the origin. So let's call this anti derivative capital P of X. So let's look at what happens at the origin me at the origin capital P at the origin is equal to. So when we plug in X equals zero into this polynomial over here, we remember that this polynomial has no cost in terms. So in this economy, we're just adding powers of X setting X equals zero shows us that this Ponyo zero at the origin. So what we're left with is PR zero is equal to the constant that we choose. Now. If we want our derivative across the origin, it must be that C equals zero. So what we have here is that there is exactly one choice of sea, which allows us to cross the Oregon, which means that there is exactly one anti derivative which crosses the origin.