1
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2
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We're asked to factor this expression by grouping.

3
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Now, they mention grouping, we're going to see what

4
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grouping is, but we're going to see very quickly that we

5
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have to do this thing called grouping because you can't

6
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just factor this expression.

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If you look at these, each of the terms, all but one of them

8
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is divisible by 5.

9
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So you can't just factor out a 5.

10
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Not all of them are divisible by either r or s.

11
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This is only divisible by r, that's only divisible by s,

12
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that's divisible by neither.

13
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So there is no common factor across all four of these

14
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terms. That's why we have to group them into groups where

15
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there are common factors, and then see if that simplifies

16
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the whole thing.

17
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And there is a little bit of an art to recognizing when you

18
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can factor by grouping, but they've set this problem up

19
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nicely for us.

20
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So if you look at these first two terms right here.

21
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You have a 5rs and a 25r.

22
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These two guys clearly have some common factors.

23
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They're both divisible by 5, they're both divisible by r.

24
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So if I just wanted to factor this one out, or if I wanted

25
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to rewrite it as a product of two expressions, how

26
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could I write it?

27
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Well, I could write it as a product of 5r times-- what's

28
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5rs divided by 5r?

29
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Well, you still have an s left over, you

30
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just have an s there.

31
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Plus-- what's 25r divided by 5r?

32
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Well, 25 divided by 5 is 5, and r divided by r is just 1.

33
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So 25r over 5r is 5.

34
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So these first two terms can be factored into these two

35
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expressions.

36
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And then let's look at these second two terms. Well, they

37
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definitely have a common factor, you have a negative 3

38
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or positive 3 common to both of these.

39
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Let's just go with the negative 3.

40
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And our goal is really to factor it into a negative 3

41
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times, hopefully, something very similar to s plus 5.

42
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And you might already be seeing that it's going to

43
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factor into s plus it.

44
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5.

45
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So let's factor out that negative 3.

46
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So these two terms you can rewrite as negative 3 times--

47
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what's negative 3s divided by negative 3?

48
00:02:01,769 --> 00:02:04,509
Well, you're just going to have an s left over.

49
00:02:04,510 --> 00:02:08,219
And then what's negative 15 divided by negative 3?

50
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Well, that's just positive 5.

51
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52
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And just like that, we've grouped them and we're able to

53
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factor each of those groups, and then something interesting

54
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might pop out at you.

55
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And one, you can always verify that you factored this

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properly by distributing each of these expressions.

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Distributing the 5r times s plus 5, and the negative 3

58
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times s plus 5, you'll get exactly this.

59
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But something maybe jumped at out you just now.

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You have 5r times s plus 5.

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Then you have negative 3 times s plus 5.

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So now this expression, we have two terms instead of

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four, right, this is one term, this is another term.

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And they both have s plus 5 as a common factor, so we can now

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factor out s plus 5.

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67
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So this whole thing can be rewritten as s

68
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plus 5 times 5r.

69
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Right?

70
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If you take 5r times s plus 5, and you factor out the s plus

71
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5, you're just left with the 5r.

72
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And then similarly, if you take negative 3 times s plus

73
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5, and you factor out the s plus 5, or divide by s plus 5,

74
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you just have a negative 3, just like that.

75
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And then we're done!

76
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We've factored this expression by grouping.

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It's s plus 5 times 5r minus 3.

78
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And you can verify it by multiplying it out.

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If you distribute the s plus 5 onto each of these terms,

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you'll get this expression up here, and then if you

81
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distribute the 5r over there you're going to get that

82
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expression.

83
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If you distribute the negative 3, you're going to get that

84
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expression.

85
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So this does simplify to that, so we have factored

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