How many roots in a polynomial

There isn't a general method for finding roots, sadly. It may help to remark that a multiple root is also a root of the derivative, and it's a lot easier to find common roots of two polynomials of close degree. If so, I can write out an approach for detecting multiple roots. Beyond 4th degree polynomials there is no general formula to write down their roots.

This is known as the Abel-Ruffini theorem. Show 10 more comments. Active Oldest Votes. Does this help? Doug M Doug M 5, 1 1 gold badge 8 8 silver badges 13 13 bronze badges. Add a comment. I will definitely look at this again after getting into calculus. Edu Edu 1, 1 1 gold badge 9 9 silver badges 23 23 bronze badges. The comments clarify that the OP is curious about unique zeroes. If there are further questions motivated by the answer, anyone can ask in the comments Adam Adam 1, 1 1 gold badge 12 12 silver badges 24 24 bronze badges.

Sign up or log in Sign up using Google. Sign up using Facebook. So if the highest exponent in your polynomial is 2, it'll have two roots; if the highest exponent is 3, it'll have three roots; and so on. There's a catch: Roots of a polynomial can be real or imaginary. Mastering imaginary numbers is an entirely different topic, so for now, just remember three things:. The most versatile way of finding roots is factoring your polynomial as much as possible, and then setting each term equal to zero.

This makes a lot more sense once you've followed through a few examples. You already have the solution to the first term. If you add 4 to both sides you'll have:. Because the original polynomial was of the second degree the highest exponent was two , you know there are only two possible roots for this polynomial. You've already found them both, so all you have to do is list them:.

Here's one more example of how to find roots by factoring, using some fancy algebra along the way. A quick look at its exponents shows you that there should be four roots for this polynomial; now it's time to find them. Did you notice that this polynomial can be rewritten as the difference of squares?

The first term is, again, a difference of squares. So although you can't factor the term on the right any further, you can factor the term on the left one step more:. 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.