00:00 Good day. My name is Adedokun Afeez. I will be taking you furthermath and the topic is Polynomials.

00:22 At the end of the topic, students will be able to know how to define Polynomials. Define

00:30 a polinomia by a polinomia of lesser than degree. Use remainder theorem to find a polinomia.

00:36 Find factors of polinomias. Use roots of the cubic equation to solve a problem on cubic equations.

00:49 Now, let's define what polinomia is. An expression of the form a, n, x, roots of power n plus a,

00:56 n minus 1, x, roots of power n minus 1 plus. Obviously, when you have a, x squared, a, 2, x squared

01:04 plus a, 1, x plus a, knots is called a polinomia. So, when you have a polinomia of this form of a, a, a,

01:13 n, x, roots of power n plus a, n minus 1, x, roots of power n minus 1 plus, oops, you,

01:24 when you have a, 2, x, roots of power 2 plus a, 1, x plus a, knot is called polinomia.

01:35 We are a, a, knot, a, n, a, n minus 1, a, 2, a, 1 are called numerica constant or coefficient of x, n, x, n minus 1,

01:49 x squared and so on and so on and so on like that. Always know that in polinomia, your a, knots must not be

01:55 equal to 0, then the IS power of variable is n. Meaning that the IS power of, or the high power of unknown variable a,

02:06 is going to be n and it is called the degree of polinomia. So, if we have, for example, if we have 2 x squared

02:14 plus, plus 8, plus 8x plus 2, because an example of polinomia, the highest degree a is 2,

02:25 and the IS degree of polinomia is 2, because that is the highest power of unknown, which is 2.

02:31 So, example are, when we have p of x, so we always denotes our polinomia to be p of x equals this.

02:42 So, we only denotes our polinomia to be p of x equals this. So, always in this of knots, we're going to

02:50 be denoting our, our, our polinomia to be p of x. So, when we have p of x equals 3 x squared minus 5 x plus

02:59 2, you can see that the degree of this polinomia is 2, because the highest power of unknown variable is 2.

03:07 The second one, we have 3 x raised to power 4 minus 2 x plus 2, is an example of polinomia as well.

03:14 So, and the highest power here is 4. Then also, the following are not polinomia. So, whenever you

03:21 have f of x equals square root of 3 x plus 4 minus 2 x plus 2, is not in polinomia. So, when you

03:28 have a fraction, polinomia 2 is not in polinomia as well, that is f of x equals 3 x plus 4 minus 2 x

03:35 super 3 plus 2 x plus 2, all over 2 x plus 3. So, this is the two are not polinomia. Why the first two are

03:44 polinomia? Then, division of polinomia. How to divide, how to divide polinomia? A polinomia of degree

03:54 n can be divided by another polinomia of degree n. If n is greater than or equals to n,

04:02 meaning that if you have for example, if you have the polinomia of this form,

04:06 let's say we have x plus 2 x plus 3 is also in polinomia, because you know we have

04:12 our h is not so hot, our h is not equal to 0. And if you have another polinomia, let's say this is

04:19 pure of x for example, and we also have another, let's say this is p1, we call this one p2,

04:25 then this is 2 x squared plus b x plus c equals 0 for example.

04:32 What we are saying is that what we want to divide, what we want to use to divide if polinomia

04:36 must be less than the one we want to divide. What we are saying is that a

04:42 if polinomia of degree n can be divided by another polinomia of degree n, if our n is greater than.

04:49 So, what if n, n is what we want to divide. And what we are using to divide is x plus 2.

04:59 So, we want to use x plus 2 to divide 2 x plus b x. So, p1 must be less than the degree of p1,

05:08 the degree of polynomial of p1 must be less than the degree of polynomial of votes of p2.

05:13 That's what we are trying to say. So, let's look at the example and see. We divide the

05:17 polinomia p1 of x, p1 of x is, we have p1 of x equals x squared plus 10x plus 21.

05:30 And our p2 equals x plus 7. So, this is what we want to divide. This is what we want to divide

05:38 and this is what we are trying to divide. So, you can see that this is less than the power.

05:42 The degree of polynomial is less than this one. This is 1 and this is 2.

05:47 And p1 is being divided by p2. So, it's called dividend. So, p1 is being divided by what? p2.

05:58 Then this p1 is called dividend. So, it means what we want to divide would be called year dividend.

06:06 Then p2 is called the vessel. What we want to use to divide is x plus 7 and it's called the vessel.

06:14 And the results when we divide x plus 7 by p1, then the results are going to give us

06:21 quotient. We call the results to be what quotient. And what advice left when we divide it?

06:26 Because it's possible for us to have whenever you want to just like we want to divide numbers.

06:31 It's possible for us to have a remainder. So, also in polinomia we can also have a remainder too.

06:37 So, when you have a remainder, then I mean when you have a number left, then it is called

06:44 remainder yet is also in polinomia. So, let's divide this and let's see how we can divide.

06:50 The method we're going to use to divide here is the method we're going to use to divide here

06:57 is called long division method. So, we're actually going to talk about long division method.

07:05 But let's quickly use long division method to divide this. So, we have x plus 7

07:11 and that is our p1 of x. So, we have x plus 7 divided by x squared plus 10x plus 21.

07:29 So, let's take a look to how we can divide this and the step in which we are going to take to

07:33 divide it. Now, the first number here, the first element here. We are going to use to divide the first

07:39 element here. So, we have x divided by x squared. We give us x. Then you use x to multiply what

07:46 we use to divide it. So, we have x squared again. Then the next thing is towards to subtract it.

07:51 So, when you subtract, you subtract this, this is going to give us what? It's going to give us 0.

07:56 Then you come down, you bring this down, 10x, you bring it down, 10x plus 21.

08:02 Then you also use, you also use, so let's say we want to divide. So, we divide x, divide x squared,

08:15 we have x squared and use x to multiply x, we have x squared. Then use x to multiply 7. We have

08:22 plus 7x. So, then you subtract. So, when you subtract x squared minus x squared, we give us 0.

08:30 Then 10x plus 7x will give us 3x, 3x. So, then you use, then you take this one down. So, we have plus 21.

08:44 x divided by 3x, we have 3. Then you use 3 to multiply x, we have 3x. Then you use 3 to multiply 7,

08:56 we have 21. So, you also divide this as way. So, 3x minus 3x will give us 0. You divide until you

09:05 get 0. You divide until you get a particular number. So, then 21 minus 21 will give us 0 as way.

09:13 So, you can see that this is a divisor. This is a divisor. This is a dividend.

09:24 And this is what we call quotient. What we get is called quotient. So, take note of that. So,

09:30 you can see we have a remainder. And this is a remainder. Our remainder year is 0. So, now,

09:39 this step above to divide the polynomial by polynomial is called in long descent method,

09:46 like I said. And it can also be used to determine the quotient and the remainder of 2.

09:51 So, we can also use this method to determine the quotient and the remainder. And that is,

09:56 we can find the quotient and we can also find the remainder. Then 0, 0 polynomials. What do we

10:02 call 0 polynomials? Given that polynomial px, xn, xr to power n plus that's up to when you have

10:11 x notes, the value of x for which p of x equals 0 are called 0's polynomial. What are we saying?

10:21 Our saying is that when you have, for example, we have p of x equals x plus 2, for example,

10:28 which is a polynomial, then when our p of x is equal to 0, that is, when p of x is equal to 0,

10:34 we're going to have what x plus 2 equals 0. So, whenever you have a polynomial and it equals 0,

10:41 then we call it 0's polynomial. We call it what 0 polynomials. Because the value of our x

10:49 will determine what we want to get. So, I have to go to the minus 2 here. So, minus 2 plus minus 2,

10:56 minus 2 plus 2, we give us 0. So, now, let's take a look at this example.

11:00 Example number 2, find the zeros of the polynomials p of x equals x squared minus 5x minus 5x plus 6.

11:13 So, if you have x squared minus 5x plus 6, and what we have to find is 0 polynomials. So,

11:24 it means, for us to have this 0 polynomials. So, you said let's p of x equals x squared minus 5x

11:32 plus 6. And it's a 0 polynomial is when our p of x equals 0. So, it means when p of x equals 0,

11:42 and our p of x is this. So, it means this implies that x squared minus 5x plus 6 equals 0.

11:50 So, this is the 0 polynomials. So, the next thing is just to find the value of our x just like

11:56 a normal factorization method in our correct equation to find that. So, we have this is going to be

12:02 x squared. We're going to have this is going to be x squared minus 3x minus 2x plus 6 equals 0.

12:10 So, we're going to have x squared minus 3x and minus that is 2x we're going to have minus 6

12:19 equals 0. So, this is going to be what is common x is common. So, x minus 3 what is common

12:29 minus 3 is common we are going to have x minus 3 as well equals 0. So, we're going to have

12:37 we're going to have this is plus. So, we're going to have x minus 2 then x minus 3 equals 0.

12:43 So, it means x minus 2 equals 0 or x minus 3 equals 0. So, x equals 2 or 3. So, the value of x

12:55 a is 2 or 3. So, the value of x that would make the polynomial to be 0 are 2 and 3. If we

13:03 put it x equals to 2 in this polynomial x squared minus 5x plus 6 must give you 0. If we put 3 also

13:11 in this polynomial x squared minus 5x plus 6 we must get 0. So, that is what we call 0 polynomial.

13:17 So, the 0 polynomial of these are 3 and 2. Then, let's take a look to remember

13:26 Reminder theorem. Reminder theorem. You know, we say that in our way during our division we said

13:35 we can use long division method to find remainder and quotient. So, the long division method

13:41 enable us to find not only the quotient but remainder. So, that is what we call this is what

13:48 Reminder theorem. And suppose we are given the polynomial in example number 1

13:52 above that is px pure of x equals x squared plus 10 x plus 21. And the divisor is given to be x plus

14:00 7. We can easily, you know, in our example one that we discuss, you know, we have this. We can

14:06 easily find the remainder using the long division method. And the remainder in that example number

14:10 1 gives us 0. So, it means 0 is what is the remainder. Do you understand it? So,

14:17 and the quotient, the quotient is what we get which is x plus 3. And the divisor is what we use

14:24 to divide p of x. And that is what the divisor and the dividend is pure of x. So, we should note

14:31 that. Note that if pure of x is divided by x plus 7 then the remainder is p of minus 7. That is

14:41 the value of our x. Therefore, we can easily conclude that if pure of x is divided by x plus 8.

14:47 If for example, now for example, we have our pure of x to be a x squared plus b x plus c.

14:56 Don't forget that. This is the quotient equation. And it's also a word, it polynomial. It polynomial

15:00 can, when it, when the quadratic is small, when the highest power of, when the highest power

15:06 of unknown variable is smaller than two, is greater than two. Then we can easily find, you can easily

15:12 say it is also a polynomial. But when the highest power of unknown variable is two, that is the

15:17 quadratic equation. Now, what we are saying is that let us note that our p, let us note that

15:24 and we want to use, this is p1, this is p2. We want to use x plus 8 to divide this for example.

15:34 The remainder here is going to be what? You know, we can easily find, don't forget that we said,

15:40 we have talked about zero polynomial. And that is, we equate this to be zero and our s is going

15:46 towards minus 8. Now, the value of, so it means when we want to find, when we want to find a

15:54 remainder, we can also find a remainder by putting the value of x in our dividend. When you put

16:02 the value of x in our dividend, then whatever it gives is also a polynomial. Do you understand it at all?

16:08 You know, in one of the examples that we talked about, example number two, we got three and two.

16:15 That three and two are zero polynomial. That is our polynomial equals to be zero. And the value we

16:22 get is what, and we say that when we put three or two into that equation, it's going to give us

16:27 what? It's going to give us zero. Zero we get is the remainder. We can also get a number. If you get,

16:34 for example, if you get minus one, that's by the time you put the value of x in that dividend,

16:40 then you can also get a number, you can get minus one, you can get minus two. So whatever you get,

16:45 when you put the value of x in your dividend, then that value is called remainder. And that is what

16:55 we are talking about. So then the remainder is p of minus a. And if it is divided by x minus a,

17:02 then the remainder is what is p of a. And this form the business of what is called a remainder

17:09 theorem. This is exactly what the remainder theorem is all about. And the theorem is, if the

17:14 polynomial p of x is divided by x minus a, then the remainder is p of a. The proof is the polynomial

17:20 function p of x can be written in this form. That is, we can write the polynomial in this form.

17:27 p of x equals x minus a times q of x plus r. And this is, this is a

17:35 p of x. We can write our p of x in this form, x minus a times q of x plus r.

17:48 x minus a is our divisor. q of x is our quotient. And how does the remainder? So we can easily find,

17:56 so if you find your quotient a remainder and you want to find your polynomial back, what we need

18:03 to do is to multiply your quotient by the divisor, then how do you towards the remainder? It's to give

18:09 you the polynomial, it's to give you the dividend. Now, where x minus a is divisor? q of x is the

18:17 quotient. And how is what is remainder? Then put x is equal to a. Then you know, put x is equal to

18:23 a in this equation. You know, we said, if p1 is this and p2 is x plus a, it quets p to be 0.

18:34 And when you quets p to be 0, we have x is what minus a. Then put,

18:42 put minus a in this. Okay, this is going to be plus. It's going to be minus.

18:48 So we have p2 is x minus a. Equate this to be 0 and we have x equals what a. So when you do that,

19:00 so we have, then you put back p of a in this. Wherever you see,

19:08 x, we are putting a. So a minus a, q of a, then plus r, a minus a would give us 0 times q of a

19:20 plus r. So this would give us 0 plus r and therefore p of a equals half.

19:27 So the theorem of boom is what we call and don't forget we said in our theorem.

19:35 Now our theorem we said if p, if the polynomial of p of x is divided by this,

19:39 then the remainder is what's p of a and our p of a is what is half.

19:42 Now, a special case of the remainder theorem is when p of x leaves no remainder.

19:50 When we don't have no remainder, when it's divided by x minus a, we therefore say that x minus a

19:57 is a factor of p of x. Don't forget that we said if p1 of x is this, which is our

20:04 dividend, our dividend and p2 of x is divided. So and whenever we don't have a remainder,

20:13 meaning that our remainder is 0 for example. If our remainder is what is 0,

20:18 then we therefore say that x minus a is a factor of what is a factor of p of x.

20:24 Mean that a factor of p of x, a factor of p1 of x, mean that if this factor can also be used

20:30 to divide what. It's just like when you have 4, what are the factors of 4? 1, 2 and 4 itself.

20:38 And 1, 4, 2 can be used to divide 4. 1 divided by 4 without remainder, we're going to have this.

20:44 2 divided by 4, we have 2 and 4 divided by 4, we also have 1. So these are the factors.

20:52 So it means that when you have that, so we therefore say that x minus a is a factor of p of x,

20:58 then the modified theorem is called factor theorem. And it states that if p of a equals 0,

21:08 then x minus a is a factor of p of x. What are we saying? If for example, we are given

21:16 we are given our p of x. Let's say p of x equals x plus 2 for example.

21:28 And we want to make this one. We want to say that p minus 2, you know, is to give us what?

21:35 Minus 2 plus 2 equals 0. So this x plus 2 would be a factor of any polynomial that we have.

21:44 This x plus 2 because it is only when it is when it's equal to 2 or when it's equal to minus 2

21:51 that we can get 0. It means our p minus 2 equals 0. So then x minus a is a factor of p of x.

22:01 Then example, we have given that p of x equals 3 x squared minus 4 x squared plus 2 x plus 3,

22:11 find the remainder when p of x is divided by x minus 1. We have to find the remainder when p of x

22:17 equals x minus 1 is divided by x minus 1. So what are we going to do? So we're going to equate this.

22:26 We said we have 3 x squared 3 x raise to power 3. That is our p of x and 3 x square is to power 3

22:42 minus 4 x squared plus 2 x plus 3. And we have to find the remainder. So you know that let's say

22:53 p 1 of x is our x minus 1. So when we equate this to be 0, our head is going to be 1. So

23:02 putting p putting x in p of x in this p of x that is 3 x squared minus 4 x squared plus 2 x plus 3.

23:16 So we have p1 equals 3 into 1 or is to power 3 minus 4 into 1 is to power 2 plus 2 into 1

23:27 is to power 3. So when we solve all this, when we solve all this, we're going to have 4.

23:35 You know like we say that it's not necessary that we get 0. So when we solve all this we get 4.

23:40 So our 4 year is what is remainder. So and if we use x minus 1 to divide x raise to power 3 minus

23:46 4 x squared plus 2 x plus 3. Whatever you get at the hand, when you use lung division method,

23:52 whatever you get which is the number, we are going to get full as way when we use lung division

23:57 method. So we can use lung division method to find the remainder and questions. And as

24:01 well, we can also use what we call a remainder theorem and we can also use what we call the factor

24:06 theorem to find remainder. Now I've been talking about the how we can find the vision of

24:16 Puy-Numia. I also talked about how we can find a remainder using the remainder theorem.

24:24 And now we can also find questions and remainder using our remainder theorem and now we can find

24:29 factor. We also talked about factor theorem. So the factor theorem, we also use it to find

24:34 our remainder as way. And we have talked about that. So we have this evolution that is the first

24:41 one is divide Puy-Numia P1 of x equals 2 x squared equals 2 x squared minus 3 x plus 9 by P of 2,

24:51 P 2 of x equals x plus 3. We have to use lung division method. And find the remainder when x

24:58 is equal to 3 plus 5 x squared minus 3 x plus 1 by x plus 1. We can easily find our remainder

25:07 when you have a dividend and when you have a divisor. So is that that we use? Is that that we use

25:14 a factor theorem? We use what we call remainder theorem to do that. Then find the zeros of Puy-Numia

25:22 of x squared minus 9 x plus 8. And we can, whatever we get when we have our x is the zero Puy-Numia

25:32 as we have talked about. Thank you for watching. My name is Afiz Alibuk once again. Till next time.