WEBVTT

1
00:00:00.000 --> 00:00:06.699
Here is the problem: 2x cubed minus 3x squared plus 2x minus 9 equals 0.

2
00:00:07.149 --> 00:00:10.319
A polynomial equation of degree 3 in x.

3
00:00:11.119 --> 00:00:12.628
Start with the equation.

4
00:00:13.078 --> 00:00:14.901
Move every term to one side.

5
00:00:15.351 --> 00:00:24.697
No further rational roots; approximate the remaining 3 roots numerically. The Durand–Kerner method iterates on all roots at once until they stop changing.

6
00:00:25.147 --> 00:00:26.900
Collect the real solutions.

7
00:00:27.700 --> 00:00:31.287
Now look at the graph of the left side minus the right side.

8
00:00:31.737 --> 00:00:35.313
It crosses zero exactly at the solutions we found.

9
00:00:36.113 --> 00:00:51.241
So the answer is x equals 2.0683773; complex: negative 0.28418863 plus or minus 1.4473615i.

10
00:00:51.691 --> 00:01:03.777
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.

11
00:01:04.577 --> 00:01:15.049
x equals 2.0683773: left side equals 0, right side equals 0, which checks out.