Hello everyone, I'm Susi and welcome to my channel. In this video we are going to learn to divide polynomials using the Ruffini rule, so let's do it. To divide any type of polynomial, we can use the whole division, which you have in a video on this channel if you need to see it. When do we use the Ruffini rule to divide polynomials? With very specific polynomials, polynomials that meet this peculiarity. The divisor has to follow this structure, it has to have x plus or minus a, x plus or minus a, which is equal to any number. So, here we have this division, I have this polynomial between this polynomial, which is exactly a binomial. Can I do the division by applying the Ruffini rule? Yes, because it follows this structure, x minus a number, I can do Ruffini. I can do it with the whole division, of course, the whole division is valid for any type of division of polynomials. Ruffini, you can only do it if that structure follows the divisor, as I just explained. Well, already knowing this peculiarity, let's see how to divide by Ruffini. Well, first I'm going to put the dividing in order, and I'm going to put the terms it has. What is the greater degree? The greater degree is 2, the x squared, okay? Then I will have to put the term that is raised from x to 1, and the independent term that does not have x, that's why x is raised to 0. Well, I do it like a box, like this, and I'm going to put the value it has below each one, the number that accompanies each of the x. What number accompanies x squared? The 5. What number accompanies x? The minus 2. What number is the one that does not have x? It is the independent term, the 1. Once I have it in place, now I have to put a number here. This number is determined by the divisor. How do I know what number it is? The divisor I have to equal to 0 and solve the value of x. If I pass the 2 to the other side, as it is subtracting, it passes adding. Therefore, x is equal to 2. Well, let's put that root here. It is called root, that solution, it is called root not because it is a square root, but because it is a solution. Well, I have already placed my dividing, I have placed the number that I have to put here, I'm going to start doing the operations. I lower the first number without more, and I put it here. And now I multiply this by this, 2 by 5, and whatever comes out, I put it below the next number. And now I carry out this operation that is in vertical. Minus 2 plus 10, because 10, although I do not put any sign, is positive. Minus 2 plus 10, 8 plus 8. This number, whenever I have a number here, I'm going to multiply by the one in the corner, 8 by 2, 16, and the value I put it below the next number. And once I have these two numbers, I operate them, 1 and 16, 17. And since I don't have any more numbers, I stop here. The last number that has come out is the rest. And this here is my quotient, which I have to add the literal part. Well, if the degree of the polynomial that we are going to divide is 2, the quotient is going to be a degree less, it is going to be degree 1, so I have to add it to 5, I have to add x raised to 1, a degree less than the one it has. If it is raised to 2, then x raised to 1. x raised to 1. And 8 already without x. I go in order. If the first one is added to x, the next one does not have x. And this would be the quotient of my division. Therefore, if I multiply this by this and add the rest, the dividend has to come out. It is the way to check it. Let's see one more case. This division can also be done with the Ruffini's rule because the divisor follows the structure that we have said, x and a number. Well, we are going to place our polynomial. Remember, if it has a degree of 2, then I have to place them in order. x raised to 2, x raised to 1 and x raised to 0, which is the number without x. Well, I'm going to put the box and you're going to put the numbers. What number accompanies x squared? 3. What number accompanies x? There is no term with x. There is no 2x, a 5x, no, it directly goes to the independent term. Therefore, if there is not, it is put a 0. But you have to put it. The typical error in this type of exercise is that when there is no term, you skip it, you do not put it and that is an error. Because it will square everything and it will not come out well. You have to put the 0. And now, what term accompanies x 0? That is, what term does not have x? The independent term, which is plus 2, which is positive. Well, and now the number that I have to put here, I take it out of the divisor, I equal it to 0 and I already have the value of x. The plus 1 goes to the other side subtracting. Therefore, x is worth minus 1, minus 1 is the number that I have to put here and I start with my operations. With the first one, you already know, the 3 goes down directly and I start and multiply. 3 times minus 1 minus 3, I put it under the next number and I operate this operation. Minus 3 plus 0 minus 3. This number that is here, I always multiply it by that of the corner. Minus 3 times minus 1 plus 3. And now I operate this operation. Minus 3 plus 2 plus 5. Sorry, minus 3. I don't know if I said minus 3. It's plus 3 plus 2 plus 5. And I'm done, okay? That's why I put the box of my rest, which is plus 5. And here, from here I have to get the quotient. I have to add the literal part, the letters. If it has a degree of 2, my quotient will have a degree of 1. Therefore, 3 will be accompanied by an x and minus 3 will be the independent term. This is the quotient of this division. In this section C we have this division, which as you can see, can be done by Uffini. Why? Because it is x and a number. Some of you will say, no, it's a fraction. But you know that fractions are numbers, okay? So it can be done with the rule of Uffini. Let's do it. We are going to put our box and we are going to put our dividing in order. I am not going to put the marks of the degrees of the x, okay? To get used to doing it without the marks. Well, the first degree, that is, the degree that has this polynomial, is 3. Let's start with the degree 3. The degree 3, the number that accompanies the x is 1. The next would be the degree 2. The degree 2, is there a degree 2? No, because you have to put 0. Degree 1 is that it has x. Is there a term with x? No, because we also put a 0. And the next would be the independent term, the number, the number without any x. Is there a number without any x? Yes, one eighth. You see? Well, I've already put it. And now, to get the number out of here, you know, equal the divisor to 0 and there is the value of x. In one half I pass it to the other side and it happens to me as a negative. Well, this is the number that I have to put here. Come on, it complicates us a bit. There is a fraction. I have put an easy case for you, okay? Don't worry. You know, first number I lower it directly and I start to multiply 1 by minus one half, minus one half. I do this operation, minus one half and 0 minus one half. When I get here this solution, I multiply it by what is in the corner. Minus one half by minus one half, multiplication of fractions. Minus by minus is more, that is, I already know that it is positive. I multiply numerator by numerator, 1 by 1, 1. And denominator by denominator, 2 by 2, 4. To a quarter I add 0 or the rest 0, I have the same, plus a quarter. And when I have a solution here, I multiply it by the number in the corner. Minus by plus, minus. Numerator by numerator, 1 by 1, 1. Denominator by denominator, 4 by 2, 8. And the minus that we had put before. Well, one eighth minus one eighth, 0. Come on, this division is exact. This is the rest and this is the quotient to which I have to add the letters, the literal part. The degree was 3, therefore the quotient goes to a degree minus, it is degree 2. Therefore, I am adding 1 x squared, which as it is a 1, you can not put it. The next, after the x squared, that x comes, the x. And the next is the independent term, the term without x. Therefore, this is the quotient, okay? And so far today's video. If you liked the video, give it a like and share it. Subscribe to this channel and follow me on Instagram if you want to be aware of new videos and exercises. Have a good day and see you in the next video.