00:02 Hello.

00:16 I'm Oyase Sunday

00:18 Today I'll be taking you mathematics,

00:20 the Junior Secondary School, JSS-3.

00:23 And today we'll be looking at an interesting topic.

00:26 Let's say, factorization.

00:29 Algebraic processes, factorization, algebraic processes.

00:33 Factorization has to be the big word

00:35 when it comes to algebra.

00:37 You know, ogybic express, algebraic equation, algebraic expression

00:42 are factorized.

00:44 So, now, but we'll be restricting it

00:47 because it's on the broader side,

00:49 so we'll be restricting it to what we're expected to talk about.

00:52 Okay, the objective is to be able to factorize

00:54 simple ogybic expression.

00:56 Okay, simple ogybic expression,

00:58 factorize it as simple as possible.

01:01 Factorize quadratic ogybic expression.

01:05 That is expression that takes the square.

01:08 The simple ogybic expression

01:09 and the expression without the square.

01:11 Now, the factorization of the ogybic,

01:14 quadratic ogybic now becomes the ogybic

01:17 that has a square.

01:19 Okay, so we have solving what problem using factorization.

01:24 Those are the objective for the code.

01:26 So, we have factorization can be seen from two perspective,

01:31 factorization can be seen from two perspective.

01:34 You write a number or an expression

01:36 as a product of itself.

01:38 When you write a number or an expression

01:40 as a product of itself is factorization.

01:43 You write a number.

01:44 We're going to be solving questions on it

01:46 or first, we're going to lay down the rules.

01:49 Our factorization, you write a number

01:51 as a product of itself.

01:53 Now, you could also arrange terms

01:55 in a more convenient way.

01:57 Now, while you are already terming a more convenient way

01:59 or you write the terms in two forms,

02:02 your code you are going out with what?

02:04 Factorization.

02:05 Now, good and fine.

02:07 Factorizing is simple ogybic expression.

02:10 Let's start with that.

02:11 The first question says factorize.

02:13 Why are we using the word factorize?

02:15 Because we are having a combination of alphabet and numbers

02:19 and signs are in the two.

02:20 So, when the sign comes in, it's factorization.

02:24 That is, if you are grouping them to the way they belong

02:27 or you are trying to collect them together.

02:29 So, factorization, we have factorize 5xy plus 10x.

02:36 Now, from the word factorization comes factor.

02:40 Facto means what are the things that distance middle up off?

02:43 So, let's factorize, picking the first question.

02:45 Now, I have taken out factorization.

02:50 Simple one, starting with the simple one

02:53 when we move up the ladder.

02:55 Now, factorize.

02:56 A factorizing 5xy plus 10x.

03:04 First, pick the first one.

03:06 Let's pick the first one.

03:07 We have 5xy, what are the elements

03:11 or what are the factors that mix up 5xy?

03:14 Now, the factors that make up 5xy

03:17 include 1, 5, 2 factor that mix up is 2.

03:25 1 is 5 multiplied by 2 is x and 3 is the y.

03:32 Now, look at it carefully.

03:34 These three factors, they came together to mix up 5x.

03:39 So, they could also be split apart.

03:42 Now, picking the second one, 10x.

03:45 Now, you have 10x.

03:47 Picking the second one, 10x.

03:48 I have 10x.

03:49 What are the factors that mix up 10x?

03:51 10.

03:53 Now, what before you go into 10?

03:56 Before you go into 10, there's something that is very important.

03:58 Before you go into 10 is, now, before I take out 10,

04:03 what are the possible numbers that could multiply 10?

04:07 Effectively, without the remainder.

04:10 10 itself, yes?

04:12 2 itself, 2.

04:15 3 cannot multiply with 3.

04:16 If we divide 10, there will be a remainder.

04:18 2 divides 10, there's no remainder.

04:21 4 divides 10, there will be a remainder.

04:23 5 divides 10, there's no remainder.

04:25 So, we have 2 and we have 5, and we have 10 itself.

04:30 So, you pick up the possible factors.

04:31 So, we have 2 times 5, then times x.

04:38 So, when we want to factorize this, we will simply say,

04:41 what's common to both sides?

04:43 What's common?

04:45 What's common to do?

04:46 Looking at this, what's common?

04:47 You could see from the line drawn, you would see that we have 5 and x is common.

04:55 We have x-trace to x-common, and we have 5-trace to 5-common.

05:01 You can see the trace.

05:03 Now, looking at this trace carefully, so pick these two out.

05:06 So, we could say our final answer is 5 times x.

05:12 That's 5x, good.

05:16 Now, you say, oh, oh, oh, oh, oh, okay, so we'll say, yes, that's it.

05:21 So, I said, observe that the two terms have 5 and they have x,

05:25 and x becomes their common factors.

05:28 So, 5x plus 10x will have the common factor to be 5 multiplied by y.

05:35 So, therefore, now, we pick up with another one, another question.

05:40 Now, still repeating the same process, still repeating the same process,

05:44 but if you look at this question carefully, you observe that we have 12.

05:48 You pick up the common factors of 12.

05:50 What are the values, numbers that can divide 12 without no remainder?

05:54 We have 2, 3, and for 15, we have 3 and 5.

06:00 Now, when you pick it out like that, let's pick it out so we could have it carefully.

06:06 Now, look at it. I have 12. I have 12 a square b.

06:15 Now, 12 a square b, what are the common factors for 12?

06:19 The common factors for 12 are what?

06:21 They are 2 and 2 multiplied by 2.

06:25 They're multiplied by 3 multiplied by 3.

06:30 The question is, why am I multiplying it by 3?

06:31 Because a 2 multiplied by a 2 gives 4.

06:36 A 4 multiplied by a 3 gives 12.

06:39 Then, I now have multiplied by...

06:46 multiplied by what?

06:49 A multiplied by A.

06:53 First, take it out.

06:54 To have 12 common factors in bigger space for it.

06:58 We'll have 2 multiplied by 2, then multiplied by 3.

07:05 Like 2 multiplied by 3 multiplied by a square, that means a is 2.

07:10 By a multiplied by a multiplied by a b.

07:16 Now, we go for the second one, that's 15.

07:18 I have 15.

07:19 What are the possible factors of 15?

07:22 I know the ba is the square.

07:24 What are the possible factors of 15?

07:26 I have 3 multiplied by a 5, which gives us 15.

07:31 So 3 times 5 is 15 multiplied by...

07:34 How many portions do we have of a?

07:36 1 multiplied by the b is 2, that's square.

07:39 b times or b.

07:42 Now, if we look at it carefully, expansion comes in place.

07:48 Look at it carefully, works and say, okay, let's look at it.

07:51 Let's trace it out.

07:53 What would be the possible common factors between these things?

07:56 The possible common factors that can pick up, pick out themselves.

07:59 Looking at it carefully, what do we have?

08:01 We have 3, top up, a 3.

08:05 We have 2, there are no 2, 2, there are no 2.

08:09 We have 3, top up, a 3, a, a, top up, and a.

08:15 Now, do we have another one, a, this a, no, no other than b, take up a b.

08:21 So picking out this out of the question, what do we have?

08:27 We're not going to have it as what?

08:30 A half, I'm going to take out, we're taking out 3, we take 3 multiplied by, what do we have?

08:37 A multiplied by, what do we have b?

08:40 So if we take it out, we're going to be having 3, a, b.

08:45 Now, 3, a, b, 3, a, b, 3, a, b becomes much of a common.

08:52 But wait, 3, a, b is common to the both sides.

08:59 But that now, the final answer, this is common to both sides.

09:02 Now, how do we now relate it to doing a two of them?

09:06 Now, you now go back to your question and say, 3, a, b, 3, a, b, open bracket.

09:15 How many of the 3 a b's can be found in 12?

09:17 How many 3 can you find in 12?

09:19 4, a, take out a, b, take out b, what is left?

09:25 We have left with what?

09:27 A, because we have a squared plus, we have 3 into 15, I have 5.

09:35 Then I have a, take out a, we have 2 b.

09:39 1 b, take out 1 b, we will let it be.

09:41 So we have it in this point.

09:42 So we could have that as our final answer.

09:46 If you multiply this, it will, it turns back.

09:49 That's factorization.

09:51 Now, with this, that giving you a wider idea on how to solve

09:57 questions that involve what factorization.

10:00 Now, on this, you could have 10 square, 10 square,

10:04 18 square, 13 is to power 3, 14 is to power 4.

10:07 Simply follow this step by step through, like I did in the second one.

10:11 It will give you a broader idea on how to solve

10:14 the question categorically.

10:16 Well, going further, there's a need to observe

10:19 that in these three, if you look at these three carefully,

10:22 first of all, find out what, what can make break each of them.

10:26 In factorization, you are expected to break.

10:28 Break them into pieces.

10:31 Then, try to bring out the pieces together

10:33 to see if they could go up.

10:35 Now, categorically move up, move up the circle.

10:38 We have, picking it, then the other example.

10:41 Let's take another example for better understanding.

10:44 I have what?

10:46 I have 10 x square.

10:50 What are the possible factors of 10?

10:53 I have 2 multiplied by 5 multiplied by how many x?

10:58 Multiply by x multiplied by x.

11:00 I have what?

11:01 18.

11:03 I have 18 x with to power 3.

11:06 That means there are 3 x.

11:08 Now, what are the possible values?

11:09 I have 2 multiplied by 3 multiplied by another 3.

11:18 Now, how many x do we have?

11:19 3 x multiplied by x multiplied by x multiplied by x.

11:25 Now, I have what?

11:26 14.

11:27 What are the possible factors of 14?

11:29 We have to power what?

11:30 4.

11:31 What are the possible factors of 14?

11:32 I have 2 multiplied by 7 multiplied by x multiplied

11:38 by x multiplied by x for x multiplied by x.

11:43 Taking it out, what's common in the 3 process?

11:48 In the 3 process, if you are very observant, you will see 2, 2 connecting to this and 2 connecting

11:53 to this.

11:54 They make it more easier.

11:55 You will see an x, yeah, connecting to this and connecting to this.

12:01 Make it more easier.

12:02 You will see another value of x.

12:06 Yeah, can connect to this and this can connect to this.

12:11 So, looking at it carefully, picking those 3 out of them, what do you observe?

12:17 You observe carefully that you have 2 multiplied by x multiplied by x.

12:25 2 multiplied by x multiplied by another x.

12:30 So that we will have, what we have, 2x squared, good.

12:38 2x squared is common to the 3 of them.

12:41 That gives us an example to be able to factorize them out.

12:44 Now, the question is, how do you now rewrite them?

12:47 How?

12:48 How do you write the question out?

12:49 You pick it out and say, OK, pick out, pick out 2x squared.

12:53 Close it.

12:54 Open a bracket.

12:55 2 into 10, that gives 5.

13:00 x squared, x squared.

13:03 What will be left there?

13:04 Nothing.

13:05 x squared, take out x squared.

13:06 2 into 19.

13:10 That gives us 9.

13:12 x squared, taking out x to the power 3, will be left with 1 x.

13:16 Then I have plus 2 into 14 is 7 and I have x squared, taking out x to the power 4, will

13:24 be left with 1, 2x.

13:27 Oh, beautiful.

13:28 Now, this has given us a better idea on the factorization process.

13:33 Which gives you wider view to be able to solve this.

13:37 Now, there's another one that is more careful, factorization of quadratic.

13:41 We've looked at factorization of the simple 10, simple algebraic expression, but there's

13:47 factorization of quadratic equation.

13:52 Now, that is an I have a show of a factorization, after the simple or move to the quadratic.

13:58 Now, in the quadratic, what do you do?

14:01 This is where the values or the unknown has a square.

14:07 That means it has a power of 2, but look at that.

14:13 We have three identities in what, in quadratic, a plus a square gives a square plus 2 a b.

14:26 Plus b squared.

14:28 How does that work?

14:29 Someone says, how does this work?

14:31 This simply works in this way.

14:33 I have a plus a square.

14:38 A square here means the bracket is in 2, so I have 2 brackets.

14:45 I have a plus a, open bracket a plus a.

14:52 Now, that's what it means, a square.

14:55 So I have 2.

14:57 So instead of being right in the 2, I could just bring them and put the square.

15:01 So when I see the square, I know that.

15:02 So this means 2 bracket.

15:05 And in bracket under factorization, when you see 2 brackets, you use every element in

15:10 the first bracket to open the second bracket.

15:14 You put it again.

15:15 In factorization, when you come across 2 brackets, use every element in the first bracket to open

15:22 the second bracket.

15:24 So now, let's go by solving this.

15:26 So I have, what is the first element we have is a.

15:30 The first element is a, open bracket a plus b, close the bracket.

15:37 I want the second element plus b, open bracket a plus b, close the bracket r.

15:44 Now, what do I do next?

15:46 Open the bracket.

15:46 Remember, multiplication comes.

15:48 A times a gives me a square plus a times b gives me a and b.

15:56 And plus times plus, I have it as addition.

16:01 b times b gives me b square because they are 2 multiplying themselves.

16:08 So I have a square plus a b plus a b.

16:13 They are 2.

16:14 We are crossing them together.

16:15 That gives me 2 a b plus b square, 2 a b plus b square.

16:23 Now, that's the perfect square, you could see, a square plus 2 a b plus b square, that's

16:34 the perfect square.

16:36 So with that, I will give you a exercise to do the expansion.

16:40 That's opening of bracket for the second and for the third.

16:45 So I've been able to open the bracket, this means they are 2.

16:48 So a minus b, a minus b.

16:50 Use the first bracket to open the second bracket.

16:52 It becomes easier equation.

16:54 Now, that takes us to solving of quadratic.

16:57 How do we solve quadratic expression?

17:00 If we have factorization in quadratic.

17:02 Now, we have to say, observe the expression and the first and the third term are perfect.

17:07 If the first term is x square, and the last term is a square, then it's more easier to

17:13 express in quadratic, factorize each of the term, then it's better.

17:18 Now, how do we make that possible?

17:20 Someone would say, how do we make that possible?

17:22 Now, the first thing is, let's pick the question.

17:26 We have the question is x square plus 8x plus 16.

17:35 Take note of this and take note of this, very, very important.

17:41 When you're solving, take note of the last and the first.

17:47 This is already in square, good.

17:49 But can this be expressing square?

17:51 Does it have a square?

17:52 It's can you go in a square root?

17:54 Or we have square root, 4, note of 4, 2, get it.

17:59 Now, square root of 9, 3, note of 16, 4, note of 25, 5.

18:06 So if these are in a perfect square, it's easy, very easy.

18:12 So you put it in square and you express this how, how do you break it out?

18:17 So someone says, oh, it becomes simpler.

18:20 So I have a solution to the problem.

18:24 Now, this gives me x square plus 2, 2, 2, 2, multiply by x, multiply by 4.

18:36 I'm breaking it into 10 plus this is a perfect square.

18:39 I will have 4 less to power 2.

18:42 Because 4 less to power 2 gives me 16.

18:44 That's 4 in 2 places.

18:46 Remember I was talking about perfect square, yeah?

18:48 And I have perfect square means I can take it up, split it out, and we turn it back.

18:53 It will mean where they are, too.

18:55 I have 4 less to power 2, 4 times 4, 16.

18:59 It goes back to 16.

19:01 4 square 2 goes back.

19:03 So, but if it's not a perfect square, then you take it out.

19:06 But now, it's a perfect square.

19:08 So it's easier to go in that line.

19:10 So you have your 4, 2 square plus 4 square.

19:16 Now, looking at it vividly, you will see s square, you will see a 2, and you will see a x.

19:22 Now, the next question, the next thing you look at from the question is, how do I group this?

19:29 I need to group them into 2, 2, 2.

19:32 So I could easily find what's common among them.

19:36 I need to group them into 2, 2.

19:38 Because someone would say, OK, what do we need to do?

19:40 Now, taking it cardioically, we'll move it a step.

19:44 I will have it as what?

19:46 x x square plus 4 x, meaning they are square square.

19:55 So I could have, because of the square, I could have there as 4 plus x plus 4, close the bracket,

20:01 and x plus 4, close the bracket.

20:05 It means that they are 2.

20:07 These 2.

20:08 So x square plus 8x plus 16 would simply give me x plus 4, close the bracket, x plus 4, close the bracket,

20:23 meaning they become perfect square.

20:26 Taking the square, going back to the law.

20:29 Taking the square of the first, the square of the second, putting them out the bracket,

20:33 become more simpler to do.

20:35 Now, this creates a very easy expression to go.

20:39 But you could also take it the other way, by having x square plus 8x plus 16, and we could

20:46 have 4x plus 4x plus 16, the 2 other ways could go.

20:51 So I could have it, tipping it in into the expression, or you could go in this way, say, oh,

20:58 well, if I want to go the other way, the other way, you could come up, or put it down,

21:04 or you could take it as x square plus 4x plus 4x plus 16.

21:11 Now, what have we done?

21:12 You split it into 4.

21:14 So these 2 and these 2 go.

21:17 So we have between the and near ones, come on, break out, bring out the x.

21:20 So we could have it as x, open the bracket, x plus 4.

21:26 Between the and near ones, come on, we have plus 4, 4 can go.

21:29 So we have, open the bracket, I have categorically 4 plus 4, I have x plus 4.

21:38 Now, good.

21:39 Why?

21:40 Now, what am I doing in this way?

21:42 You could either go this square, or you could go this spot.

21:45 You're still going to get, arrive at the same answer, 2 directions, or giving you same

21:51 answer.

21:52 So I could say, OK, between the and near, what is common?

21:55 Pick the ones.

21:56 The 2 brackets will be the same.

21:57 So you pick it up x plus 4, the one by the side, and I have x plus 4, I conclude.

22:03 We done.

22:04 Now, the last part of the factorization process is word problem.

22:11 They're not giving an expression.

22:13 Given the word form, how do you pick it out?

22:15 Now, this is actually what they're talking about.

22:18 Now, now, that could be a bit difficult.

22:21 I've seen a lot of students pick that question.

22:24 And after we look at it for a long period of time, don't know where to start from.

22:29 Now, the simple, it has a one simple rule.

22:33 The one simple rule is, do it through first.

22:36 Pick out what you expected to do.

22:38 Then form it as a whole and look at it in a word problem.

22:41 I have, in a certain school, there are twice as many boys as there are girls.

22:49 There are twice as many boys as there are girls.

22:52 Now, every word problem gives you an insight to what you want to do.

22:56 But technique, that means in that school, there are boys and there are girls.

23:00 But there are twice as many boys as there are girls.

23:05 They are trying to point the direction to the boys.

23:07 Now, the total number of students is 360.

23:11 How many girls are at the school at that particular time?

23:15 The first thing is, the question here is trying to introduce that the numbers,

23:21 there are twice as many boys.

23:24 So the multiplication means what is the number of the girls first?

23:28 Then we can now get the numbers of the boys through the numbers of the girls.

23:34 Reverse, verse.

23:36 So, in what problem?

23:37 Pick out the variables.

23:39 The variables here are the boy and the girl.

23:42 So, let's go.

23:44 I'll pick it out.

23:45 Pick, you say, the boy and the girl.

23:48 You just say, let the boy, let the girl, let the girl be X.

23:57 You can use any value.

23:59 You can be let the girl, the number of girl, BX, PQ, be any alphabet to pick, BX.

24:07 We're told that the twice, the boys are twice that of girls.

24:14 So, boys are twice that of girls.

24:16 So, we'll just pick it up and say, let the numbers of boys now at the given in the question

24:25 be twice of that of girls.

24:28 And what's girls?

24:29 X.

24:30 So, it would be two times X.

24:32 That gives me word, two X.

24:34 Someone said, oh, are you manipulating it out?

24:37 I'm following street instruction.

24:39 In what problem?

24:40 Don't manipulate with your own.

24:43 Follow the street instruction.

24:45 Don't try to play around, follow the street instruction.

24:49 From X strat, you can now move, but follow the street instruction.

24:54 Two X.

24:55 Now, next says, the total number of students in that school is 360.

25:00 So, I'm not trying to look at amenable of girls, total number of students.

25:03 So, I'm talking about the song.

25:05 I'm summing, total number of students.

25:07 That means, our boys, plus girls in this school, must be equal to 360.

25:15 From the extra.

25:17 Now, what are the number of boys?

25:19 We have it as two, two X plus what's the number of girls?

25:22 We have it as what?

25:23 X.

25:24 Equal to 360.

25:25 So, I have two X plus X.

25:28 Remember?

25:29 That gives me word, three X equal to 360.

25:32 Now, I have, finding them, I have dividing both sides by three.

25:39 The dividing both sides by three, divide both sides by three, three, three, take out three,

25:42 a one, two, a, will give us what?

25:46 120.

25:47 So, the number of girls is 120.

25:54 And someone said, okay, the number of girls are, the number of girls in this school is 120.

26:00 Now, the question is, what would not be the number of boys?

26:04 It would be twice the number of girls, so that would be 120 times two.

26:08 So, 120 times two, that would give us 240.

26:11 So, when you had 240 plus 120, it would give you 360, total number of students in the school.

26:18 What problem?

26:19 So, the way it is.

26:20 Now, with that, it's not able to give you an idea of what to do on the next question,

26:26 it would give you an idea of what to do.

26:27 Now, move to the evaluation.

26:30 To test your knowledge or how far you've understood what you've been taught.

26:34 We have question number one, said evaluate.

26:37 A square plus 8A plus 16.

26:41 That's factorized, you're factorizing it.

26:44 A square plus 8A plus 16.

26:46 Factorized, P square minus 10P plus 25.

26:52 Factorized, P square minus 10P plus 25.

26:57 The last one says factorized, x square plus xy plus 8x plus y plus 8y plus y plus 8y.

27:06 So, you could see that I had to bring it into four.

27:09 So, you could group it into two, two, two.

27:11 Factorized, a square plus xy plus 8y plus 8x.

27:21 So, we are engineering it anyway, but notes you have to rearrange before you could solve.

27:27 Very important.

27:28 Thank you very much.

27:29 I've come to the end of the class for today.

27:31 My name remains, or yes, it was a Sunday, see you in the next class.