1
00:00:01,240 --> 00:00:02,720
NARRATOR: Exercise six.

2
00:00:02,720 --> 00:00:10,120
Solve the equation x cubed +
2x squared + 3x plus 6 equals 0.

3
00:00:10,120 --> 00:00:12,480
So we first need to factorise
the left-hand side

4
00:00:12,480 --> 00:00:15,120
in order to be able to use
the null factor law,

5
00:00:15,120 --> 00:00:19,440
and in factorising a cubic,
we need to first identify one factor.

6
00:00:19,440 --> 00:00:24,200
And we do that by looking for
one root of this equation -

7
00:00:24,200 --> 00:00:28,080
so we're looking for a number that
we can substitute into the polynomial

8
00:00:28,080 --> 00:00:29,920
in order to make it 0.

9
00:00:29,920 --> 00:00:35,400
So I'm going to start with
the value of the polynomial at 1.

10
00:00:35,400 --> 00:00:40,840
And that's gonna give me
1 + 2 + 3 + 6

11
00:00:40,840 --> 00:00:43,400
which is in fact 12, so not 0.

12
00:00:43,400 --> 00:00:46,200
So therefore this graph goes through
the point (1, 12).

13
00:00:46,200 --> 00:00:47,840
That point isn't an x-intercept

14
00:00:47,840 --> 00:00:50,600
and that's not useful to us
in the factorisation.

15
00:00:50,600 --> 00:00:52,360
I'm also noticing from this

16
00:00:52,360 --> 00:00:55,320
that all the terms
in my polynomial are positive,

17
00:00:55,320 --> 00:00:58,080
and so I'm gonna need to be
substituting in a negative number

18
00:00:58,080 --> 00:00:59,800
in order to get 0.

19
00:00:59,800 --> 00:01:02,760
I also know that
the numbers I substitute in

20
00:01:02,760 --> 00:01:04,280
are gonna all have to be ...

21
00:01:04,280 --> 00:01:07,960
... if they're going to give me 0,
are going to need to be factors of 6

22
00:01:07,960 --> 00:01:11,040
because the only way the brackets
can expand out

23
00:01:11,040 --> 00:01:14,480
to give a constant term of 6
at the end

24
00:01:14,480 --> 00:01:17,200
is using factors of 6
in those brackets.

25
00:01:17,200 --> 00:01:20,760
So let's first have
a look at P at –1.

26
00:01:20,760 --> 00:01:26,200
So that's gonna give me
-1 + 2 – 3 + 6,

27
00:01:26,200 --> 00:01:31,200
which gives me 4,
which, again, isn't 0.

28
00:01:31,200 --> 00:01:33,920
So we know we need to
stick with the negative,

29
00:01:33,920 --> 00:01:35,520
so we'll try –2.

30
00:01:35,520 --> 00:01:40,480
That gives me –8 + 8 – 6 and + 6

31
00:01:40,480 --> 00:01:42,280
which is indeed 0.

32
00:01:42,280 --> 00:01:48,040
So that tells me that
(x + 2) is a factor

33
00:01:48,040 --> 00:01:52,880
in order to give me one of
my x-intercepts at x = –2.

34
00:01:52,880 --> 00:01:56,240
So now that
I've identified one factor,

35
00:01:56,240 --> 00:01:59,400
I need to divide my cubic
by that factor

36
00:01:59,400 --> 00:02:01,560
to determine the other factors.

37
00:02:01,560 --> 00:02:04,920
So I use long division
for this process.

38
00:02:04,920 --> 00:02:07,040
There are different ways
to go about doing this.

39
00:02:07,040 --> 00:02:08,520
There's synthetic division.

40
00:02:08,520 --> 00:02:13,160
You can also do it in
a sort of more linear fashion

41
00:02:13,160 --> 00:02:15,480
without actually writing out
any kind of division.

42
00:02:15,480 --> 00:02:18,600
At the end of the day, you do
all the same mathematical steps

43
00:02:18,600 --> 00:02:20,160
to determine the factor.

44
00:02:20,160 --> 00:02:23,040
It's just about what
you choose to write down.

45
00:02:23,040 --> 00:02:27,160
So I'm dividing the cubic
by the linear factor,

46
00:02:27,160 --> 00:02:31,920
so what we're looking at here
is we've got our cubic equation

47
00:02:31,920 --> 00:02:36,600
x cubed + 2x squared + 3x + 6,

48
00:02:36,600 --> 00:02:39,360
and we know that
that's equal to (x + 2)

49
00:02:39,360 --> 00:02:42,120
multiplied by some other factor.

50
00:02:42,120 --> 00:02:44,280
And obviously we can see

51
00:02:44,280 --> 00:02:48,600
from this linear
right up here on the right

52
00:02:48,600 --> 00:02:50,440
that the first term
in that bracket there

53
00:02:50,440 --> 00:02:52,280
is going to have to be x squared.

54
00:02:52,280 --> 00:02:54,520
And what we're doing to get that
x squared is we're asking ourselves,

55
00:02:54,520 --> 00:02:57,600
"Well, what would I need to multiply 
x by in order to get x cubed?"

56
00:02:57,600 --> 00:02:59,840
And that's what we do
in the long division as well.

57
00:02:59,840 --> 00:03:02,400
"What would I need to multiply x by
to get x cubed?"

58
00:03:02,400 --> 00:03:04,160
Well, that's x squared.

59
00:03:05,320 --> 00:03:06,960
And then I'm gonna expand that out,

60
00:03:06,960 --> 00:03:09,240
just as we would when
we're expanding out the brackets -

61
00:03:09,240 --> 00:03:12,400
we would do x times x squared
and also 2 times x squared.

62
00:03:12,400 --> 00:03:15,680
So we're gonna do
x squared times (x + 2)

63
00:03:15,680 --> 00:03:19,120
which is gonna give me
x cubed + 2x squared.

64
00:03:19,120 --> 00:03:21,080
And then we subtract this.

65
00:03:21,080 --> 00:03:26,520
And we've chosen x squared in order
that the x cubed will cancel out,

66
00:03:26,520 --> 00:03:29,320
and in fact we have
an interesting situation here

67
00:03:29,320 --> 00:03:31,920
where we've also got the 2x squareds
cancelling out.

68
00:03:31,920 --> 00:03:35,080
So 2x squared – 2x squared
gives me 0.

69
00:03:35,080 --> 00:03:36,720
And I'm gonna write "x squared" here

70
00:03:36,720 --> 00:03:39,520
just to maintain the sort of
place value of my columns.

71
00:03:39,520 --> 00:03:42,240
Then I bring down the next term,
+ 3x,

72
00:03:42,240 --> 00:03:43,800
and we repeat the process.

73
00:03:43,800 --> 00:03:46,920
So what would I need to multiply x by
to get 0x squared?

74
00:03:46,920 --> 00:03:48,400
Well, that would be 0x,

75
00:03:48,400 --> 00:03:51,240
so we're not going to have
an x term in our factor.

76
00:03:51,240 --> 00:03:52,800
And then we expand out.

77
00:03:52,800 --> 00:03:55,680
So 0x times 0x, that's 0x squared.

78
00:03:55,680 --> 00:03:58,680
0x times 2, 0x.

79
00:03:58,680 --> 00:04:01,280
Then we take away and we've got 3x.

80
00:04:01,280 --> 00:04:02,800
Bring down the next term,

81
00:04:02,800 --> 00:04:04,280
which is + 6.

82
00:04:04,280 --> 00:04:05,680
What will we need to

83
00:04:05,680 --> 00:04:07,200
multiply x by to get 3x?

84
00:04:07,200 --> 00:04:08,760
Well, that's 3.

85
00:04:08,760 --> 00:04:11,160
Expand out, so 3x + 6.

86
00:04:11,160 --> 00:04:13,600
Write that down here.

87
00:04:13,600 --> 00:04:16,000
And subtract, which leaves us with 0.

88
00:04:16,000 --> 00:04:17,480
And that's what we expect

89
00:04:17,480 --> 00:04:19,400
when we've divided by a factor,

90
00:04:19,400 --> 00:04:21,600
we know that we're going to get
no remainder.

91
00:04:21,600 --> 00:04:23,640
So what we've identified here is

92
00:04:23,640 --> 00:04:26,040
that x cubed + 2x squared + 3x + 6

93
00:04:26,040 --> 00:04:28,160
is equivalent to (x + 2)

94
00:04:28,160 --> 00:04:30,760
multiplied by (x squared + 3).

95
00:04:32,880 --> 00:04:36,080
So now we're looking to solve
this equation equal to 0.

96
00:04:36,080 --> 00:04:38,200
So we're looking to
solve that equal to 0.

97
00:04:38,200 --> 00:04:40,280
And we can now use
the null factor law.

98
00:04:40,280 --> 00:04:44,440
So we know that x + 2 might equal 0

99
00:04:44,440 --> 00:04:47,960
or x squared + 3 might equal 0.

100
00:04:47,960 --> 00:04:50,160
So we get x = –2,

101
00:04:50,160 --> 00:04:53,640
which is the solution that
we identified back at the beginning.

102
00:04:53,640 --> 00:04:56,680
And we also have x squared equals –3.

103
00:04:56,680 --> 00:04:59,560
So we'll be trying to square root
a negative number here,

104
00:04:59,560 --> 00:05:02,840
so we're going to get no solutions
from this factor.

105
00:05:02,840 --> 00:05:06,680
So in fact we have a cubic equation
with just one solution

106
00:05:06,680 --> 00:05:09,280
at x = –2.