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# Number of possible real roots of a polynomial

Video transcript

Voiceover:So we have a
polynomial right over here. We have a function p(x)
defined by this polynomial. It's clearly a 7th degree polynomial, and what I want to do is think about, what are the possible number of real roots for this polynomial right over here. So what are the possible
number of real roots? For example, could you have 9 real roots? And so I encourage you to pause this video and think about, what are all the possible number of real roots? So I'm assuming you've given a go at it, so the Fundamental Theorem of Algebra tells us that we are definitely
going to have 7 roots some of which, could be actually real. So we're definitely not going to have 8 or 9 or 10 real roots, at most we're going to have 7 real roots, so possible number of real roots, so possible - let me write this down - possible number of real roots. Well 7 is a possibility. If you graphed this out, it could potentially
intersect the x-axis 7 times. Now, would it be possible
to have 6 real roots? Is 6 real roots a possibility? Is this a possibility? Well, let's think about
what that would imply about the non-real complex roots. If you have 6 real, actually
let's do it this way. Let me write it this way. So real roots and then non-real, complex. The reason I'm not just saying complex is because real numbers are a subset of complex numbers, but this is being clear
that you're talking about complex numbers that are not real. So you could have 7 real roots, and then you would have no non-real roots, so this is absolutely possible. Now could you have 6 real roots, in which case that would imply that you have 1 non-real root. Well no, you can't have
this because the non-real complex roots come in
pairs, conjugate pairs, so you're always going to have an even number of non-real complex roots. So you can't just have 1,
so let's rule that out. Now what about having 5 real roots? That means that you would
have 2 non-real complex, adding up to 7, and that
of course is possible because now you have a pair here. So I think you're
starting to see a pattern. Essentially you can have
an odd number of real roots up to and including 7. So for example,this is possible and I could just keep going. I could have, let's see, 4 and 3. This is not possible because I have an odd number here. You're going to have
to have an even number of non-real complex roots. So rule that out, but
then if we go to 3 and 4, this is absolutely possible. You have two pairs of
non-real complex roots. And then we can go to 2 and 5, once again this is an odd number, these come in pairs,
so this is impossible. And then you could go to
1 real and 6 non-real. Completely possible,
this is an even number. And then finally, we could consider having 0 real and 7 non-real complex and that's not possible because these are always going to
come in pairs, so you're always going to have an even number here. So the possible number of real roots, you could have 7 real roots, 5 real roots, 3 real roots or 1 real root for this 7th degree polynomial.