Zeros of polynomials introduction

let's say that we have a polynomial P of X and we can factor it and we can put it in the form X minus 1 times X plus 2 times X minus 3 times X plus 4 and what we are concerned with are the zeros of this polynomial and you might say what is a zero of a polynomial well those are the x-values that are going to make the polynomial equal to zero so another way to think about it is for what X values is P of X going to be equal to zero or another way you can think about it is for what X values is this expression going to be equal to zero so for what X values is x minus 1 times X plus 2 times X minus 3 times X plus 4 going to be equal to 0 I encourage you to pause this video think about that a little bit before we work through it together well the key realization here is if you have the product of a bunch of expressions if any one of them is equal to 0 it doesn't matter what the others are because 0 times anything else is going to be equal to 0 so the fancy term for that is the zero product property but all it says is hey if you can find an x value that makes any one of these expressions equal to 0 well that's going to make the entire expression going to be it's going to make the entire expression equal to 0 so the zeros of this polynomial are going to be the x-values that could make X minus 1 equal to 0 so X minus 1 equals 0 we know what x value would would make that happen if X is equal to 1 if you add 1 to both sides here x equals 1 so x equals 1 is a zero of this polynomial another way to say that is P of 1 when x equals 1 then all polynomials going to be equal to 0 how do I know that well if I put a 1 in right over here this expression right if your X minus 1 that is going to be equal to 0 so you can have 0 times a bunch of other stuff which is going to be equal to 0 and so by the same idea we can figure out what the other zeros are what would make this part equal to zero what x value would make X plus two equal to zero well x equals negative two x equals negative two would make X plus two equals zero so x equals negative two is another zero of this polynomial and we could keep going what would make X minus three equal to zero well if X is equal to three that would make X minus three equal to zero and that would then make the entire expression equal to zero and then last but not least what would make X plus four equal to zero well if X is equal to negative four and just like that we have found four zeros for this polynomial when x equals one the polynomial is equal to zero when x equals negative two the bottom is equal to zero when x equals three the bottom is equal to zero and when x equals negative four the polynomial is equal to zero and one of the interesting things about the zeros of a polynomial you could actually use that to start to sketch out what the graph might look like so for example we know that this polynomial is going to take on the value zero at these zeros so let me just draw a rough sketch right over here so if this is my x axis that's my Y axis that's my Y axis and so let's see at x equals one so let me just do it this way so we have one two three and four and then you have negative one negative two negative three and then last but not least negative four we know that this polynomial P of X is going to be equal to zero at x equals one so it's going to intersect the x axis right there it's going to be equal to zero at x equals negative two so right over there at x equals three right over there and x equals negative four now we don't know exactly what the graph looks like just based on this we could try out some values on either side to figure out hey is it above the x axis or below the x axis for X values less than negative four and we can try things out like that but we know it intersects the x axis at these points so it might look something like this this is a very rough sketch it might look something like this we no without doing a little bit more work but ahead of time I took a look at what this looks like I went onto desmos and I graphed it and you can see it looks exactly as what we would expect the graph of this polynomial intersects the x axis at x equals negative 4 she let me color code it x equals negative 4 and that is that 0 right over there x equals negative 2 that's this 0 right there x equals 1 all right over there and then x equals 3 right over there in future videos we will study this in even more depth