WEBVTT

1
00:00:00.000 --> 00:00:07.047
Here is the problem: x cubed minus 4x squared minus 9x plus 36 equals 0.

2
00:00:07.497 --> 00:00:10.632
A polynomial equation of degree 3 in x.

3
00:00:11.432 --> 00:00:12.930
Start with the equation.

4
00:00:13.380 --> 00:00:15.296
Move every term to one side.

5
00:00:15.746 --> 00:00:26.683
Test candidates from the rational root theorem: x equals negative 3 is a root. Synthetic division by (x plus 3) leaves a polynomial of degree 2.

6
00:00:27.133 --> 00:00:37.954
Test candidates from the rational root theorem: x equals 3 is a root. Synthetic division by (x minus 3) leaves a polynomial of degree 1.

7
00:00:38.753 --> 00:00:48.877
Test candidates from the rational root theorem: x equals 4 is a root. Synthetic division by (x minus 4) leaves a polynomial of degree 0.

8
00:00:49.327 --> 00:00:51.010
Collect the real solutions.

9
00:00:51.810 --> 00:00:55.409
Now look at the graph of the left side minus the right side.

10
00:00:55.859 --> 00:00:59.644
It crosses zero exactly at the solutions we found.

11
00:01:00.444 --> 00:01:05.854
So the answer is x equals negative 3, x equals 3, x equals 4.

12
00:01:06.304 --> 00:01:18.309
A degree-3 polynomial has exactly 3 roots counted with multiplicity (some may be complex). Exact rational roots are found first, then the rest are solved exactly or approximated.

13
00:01:19.109 --> 00:01:26.725
x equals negative 3: left side equals 0, right side equals 0, which checks out.

14
00:01:27.175 --> 00:01:33.851
x equals 3: left side equals 0, right side equals 0, which checks out.

15
00:01:34.301 --> 00:01:40.919
x equals 4: left side equals 0, right side equals 0, which checks out.