How to solve cubic polynomials.

Welcome back to today's class. We are asked to solve for x in this polynomial. Now, for you to solve. The first thing you need to do is this cannot be easily factorized; it cannot be done like that. so we are going to use what is called trial and error in solving this, trial and error, but What do I mean by that? Let's check: if we say let x equal one, are we going to get zero? when we substitute x equals one into this equation, so we are going to have that one to the power of Three minus three times the quantity one squared minus nine times the quantity one minus five equals zero. so one to the power of three is what? One, all right? Now one squared is still one, and that one times minus three will give us minus three; now minus nine times one is still minus nine Minus five equals zero in my view when looking at this; if we solve it, you're not going to get zero. Yes, because one minus three gives us minus two; minus two minus nine gives us minus eleven. and minus eleven minus five is definitely not zero, so that means one is not a solution To this we cannot say that one is x now. Let's check what if x equals minus one. If x equals minus one, then we have minus one to the power of three minus three brackets minus one squared minus nine times minus one minus five equals zero, okay, let's see if this is going to give So what if this actually gives us zero? Now, minus one to the power of three is minus one. Now, minus one squared is one, yes, because minus times minus is plus, so minus one squared. Sorry, this is outside minus one squared, which is one. Now that one times minus three will give us minus three now minus nine times minus one is plus nine, and we have minus five here is equal zero now minus one minus three is minus four, right? Minus four now, nine minus five is what? plus four equals zero, and minus four plus four is what zero, so that means we can say That minus one is what a root of this particular polynomial, so what do you do now? So we said let's set x equal to minus one, so that means x is minus one; if it crosses the equation plus one equals zero, okay? Now, once you've done this, this is the next thing you do—we are We're going to use this to divide it so that we'll obtain a quadratic equation, okay? So let's go ahead. and use long division to divide this in order to obtain a quadratic equation, so this is what we are going to do To remember is x plus one equals zero, so we are going to use x plus one to divide. what x to the power of three minus three x squared minus nine x minus five okay so we go ahead and do that x divided by x to the power of three or x to the power of three divided by x this x this is the most important aspect of this whole thing, because this is actually what we use—we keep Using the rule for dividing powers with the same base, okay, so x cubed divided by x gives us x squared. now x to the power of two times x gives us x cubed, then x to the power of two times plus one We give us plus x squared, so whatever you obtained after division by this, you are going to use it. to multiply the two times x and plus one, then you subtract. Now when we do that, you see that x x cubed minus x cubed equals zero, so that one aside, that one has passed, so we now have minus three x squared minus plus x squared, the same thing as minus three x squared minus x squared; this minus times plus is minus, okay; now minus three x squared minus x squared will give us minus four x squared, right? So we have minus four x squared now we are We are going to bring down this minus nine x, and we are going to bring down minus five as well, all right. we keep dividing, so we are going to use x to divide minus four x squared, and when that happens we are going to have minus four x remaining, yeah, minus four x squared divided by x, so this x we Take only one of these, so we are left with minus four x, right? So we have minus four x. All right. remember that whatever you have, you are going to use it to multiply by these two terms, so minus four x times x will give us minus four x squared, okay. Now, minus four x times plus one will give us minus 4x, so we go ahead and subtract. Now when we subtract, we have minus 4x minus minus this. is zero now minus nine x minus minus four x is the same thing as minus nine x plus four x which Give us minus five x, so we have minus five x, and we are going to bring down this, this minus five. remember that what we have here is minus five x, then we bring down minus five, so we are going to Bring down minus five. This is what we have now. Okay, sorry. So this is what we have: minus five x. minus five minus five x minus five, okay. So what do you do now? We are going to use x to divide it. again as usual, now when x divides minus 5x you have what, minus 5. Now minus 5 times x will give Tell us what minus 5x and minus 5 times plus 1 will give us minus 5. Now notice that we have what we already have zero here because these two are the same, so it's zero, which means you are carry through with the division, okay? So once you get zero, sorry—this was supposed to be down here, but I know you can't see beyond this, so this is all you're going to have—zero now, that means that we have that x plus 1 times x plus 1, brackets x squared minus 4x minus 5, x squared minus 4x minus Five is equal to zero, so we have reduced this cubic to this; now it's very easy for us to solve, yes. So what do we do next? We are going to factorize this quadratic equation. We'll leave this the It's already factored, so let's factor this, okay? So you know how we do this now—you need two factors of minus 5 that when you add them or subtract them you get minus 4, okay, so um looking At this point, we have so many factors of 5, but 5 and 1 are what we can manipulate to get minus 4. Because 5 plus a plus 1 gives us 6, 5 minus 1 gives us 4, but 1 minus 5 gives us minus 4. Okay, so it's actually 1 and minus 5 that we're looking for—1 and minus 5—so for x squared. -4x - 5 = 0, we are going to write that x squared; remember that we go towards 5 times x minus 5 plus 1, right? So it's going to be minus 5x plus that one is 1x plus x. Now minus 5 is equal to 0, okay. Looking at this now, we can now factorize, looking at this and this. They have x in common, so we are going to factor out that x. x squared divided by x will leave us with x. right minus 5x divided by x we have minus 5 plus if you look at these two terms now they don't Have anything in common that means they have one in common, so one bracket x minus 5 is equal to 0 so that means that we have x minus 5, let's pick one from the brackets, x minus 5 into what x plus 1 times x plus 1 equals 0; remember, we have already found the first one as x plus 1, so it does means that three of them, x plus 1 times x minus 5 times x plus 1 again, is equal to 0, so note that since this is a cubic polynomial, that is, it has three as the highest index of your x The highest power of x is 3, which means the value of x that we are looking for must be 3. I have to get three values for x. Okay, now looking at this, you can say that x plus 1 is equal to 0. that is for this, let's also know that we have it twice, so as I'm solving for this, I'm also solving for this as well, so now we have that x is equal to this minus 1; we cross over to become minus 1, so this becomes minus 1, so we are going to write it twice: x equals minus 1 twice because of this then this is the second or the third one is what x minus 5 equals 0, that means that x is equal To what 5? So these are the values of your x. Thank you so much for watching. Do have a great day.