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Number of possible real roots of a polynomial

CCSS.Math:

Video transcript

so we have a polynomial right over here we have a function P of X defined by this polynomial it's clearly a seventh 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 nine 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 seven seven roots some of which could be actually real so we're definitely not going to have eight or nine or ten real roots at most we're going to have seven real roots so possible number of real roots so possible let me write this down possible number of real real roots well seven is a possibility this when if you graph this out it could potentially intersect the x-axis seven times now would it be possible to have six real roots is six 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 six real if you have six relax let's let's do it this way let's let's let's let me write it this way so real roots real and then non real non-real complex the reason why I'm not just saying complex is because real 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 you could have seven real roots and then you would have no non real roots and so this is absolutely possible now could you have six real roots in which case that would imply that you have one 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 you can't just have one so let's rule that out now what about what about having five real roots that means that you would have two non-real complex adding up to seven 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 seven so for example this is possible and I could just keep going I could have let's see four and three 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 three and four this is absolutely possible you have two pairs of non-real complex roots and then we could go to two and five once again this is an odd number does these come in pairs so this isn't possible and then you could go to one real and six non real completely possible this is an even number and then finally we could consider having zero real and seven non-real complex and that's not possible because these are always going to come in pair so you're always going to have an even number here so the possible number of real roots you could have seven real roots five real roots three real roots or one real root for this seventh degree polynomial