Finding zeros of polynomials
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- So I have the polynomial p of x here, and p of x is being expressed as a fourth degree polynomial times three x minus eight squared, so this would actually give you some, this would give you nine x squared and a bunch of other stuff, and then you multiply that times this. It would actually give you a sixth degree polynomial all in all, but our goal is to find the x values where that makes p of x equal to zero, or another way find the roots or the zeros of this polynomial, and in particular we're going to focus on the real zeros, the real roots of this polynomial, and like always I encourage you give a go at it, and then we'll do it together. Alright, let's tackle this. So, the way I want to solve p of x is equal to zero. I want to solve p of x is equal to zero, and figure out what, and when I say solve it I want to say well what x values will make the polynomial equal to zero. So I just need to set this right hand side to be equal to zero and then solve for x, and the best way that I can think of doing that is by factoring this out as much as I can, and if I can rewrite it as a product of a bunch of expressions equaling zero, well a product of a bunch of things equaling zero, you can make it equal zero by any one of them equaling zero, and so let's do that. So three x minus eight squared, this is already factored quite nicely, let's see if we can factor if we can factor all of this business in white, and the way I will tackle it to see if I can factor by grouping. So let me group together these first two terms, and then let me group together these second two terms, and essentially factoring by grouping is doing the distributive property in reverse twice. So from these first two terms, I could factor out, let's see what could I factor out? I could factor out a let me see, I'll just factor out an x to the third power. So I get x to the third power times three x minus eight, interesting we have a three x minus eight over there as well Now these second to two terms, I could factor out a five, so this is going to be plus five times x, sorry, times three x. Three x minus eight, very interesting, and of course I have these parentheses around all of that, and then I have three x minus 8 squared. This three x minus eight is showing up a lot, and so and of course this is going to be equal to zero. So we're gonna be equal to zero, and now I can factor out a three x minus eight over here, I could factor that three x minus eight out, and I'm going to get three x minus eight times times x to the third power, x to the third power plus five. Once again I just factor out a three x minus eight plus five, close the parentheses, and then times three x minus eight squared is all going to be equal to zero, is all equal zero. Now if what I just did looks a little like voodoo, just realize I have two terms, both of them are multiples of three x minus eight. I just factored out, I just factored out the three x minus eight. I did distributive property in reverse, so I factored it out, and what you're left with this term you just look for the next of third, and in this term you're just left with a plus five. Now three x minus eight times x to the third plus five times three x minus eight squared. Well I could just rewrite this as three x minus eight to the third power times x to the third plus five, so let me do that. So I can just rewrite this as three x minus eight to the third power, that's that times that, and then times times, do this in a nicer color, times x to the third plus five is equal to zero, is equal to zero, and now in order to get this to be equal zero, either the three x either this thing is going to be equal to zero, or this thing over here is going to be equal to zero. So let's first think about three minus eight to the third power equaling zero. So I can write this is as three x minus eight to the third power is equal to zero or x to the third plus five is equal to zero. So to make three x minus eight to the third power equal zero well that means three x minus eight is going to be equal to zero or that three x is equal to eight, divide both sides by three, x is equal to eight thirds. So that's one way to make this polynomial equal zero, x is equal to eight thirds, in fact just this right over there will become zero, zero times anything is zero. So this is a zero of our polynomial, and let's see, so x to the third, so we could say, if we subtract five from both sides, we have x to the third is equal to the is equal to negative five, and so if we take both to the one third power we could say x is equal to cube root of negative five. Now at first you might say, "Wait, can I take the square root of a negative number?" and I would say, "Of course you can!" The cube root of a negative one is negative one. The cube root of negative eight is negative two. In fact you could, even if we're dealing with reals. This is going to be a negative number. This is not going to be an imaginary number right over here, and so these are, these are the two zeros of the polynomial. There's gonna be negative, I think, negative one point something, I'm sure we could figure out it, figure out it exactly. So let's raise, so let's raise, five to the open parentheses, one divided by three close parentheses, is equal to, so that's five to the one third, so negative five to the one third power is gonna be negative one point seven one approximately. So this is approximately equal negative one point seven one. So we have two real roots, two real roots to this polynomial, or two zeros two real zeros for this polynomial, and so those are going to be the two places where we intercept the x-axis. The two x values for which where the two places where we intercept the x-axis, is the easiest way to say it.