WEBVTT

1
00:00:00.000 --> 00:00:03.808
Here is the problem: x cubed plus 1 equals 0.

2
00:00:04.258 --> 00:00:07.509
A polynomial equation of degree 3 in x.

3
00:00:08.309 --> 00:00:09.818
Start with the equation.

4
00:00:10.268 --> 00:00:12.195
Move every term to one side.

5
00:00:12.645 --> 00:00:23.210
Test candidates from the rational root theorem: x equals negative 1 is a root. Synthetic division by (x plus 1) leaves a polynomial of degree 2.

6
00:00:23.660 --> 00:00:25.343
Solve the remaining quadratic.

7
00:00:26.143 --> 00:00:27.559
Compute the discriminant.

8
00:00:28.009 --> 00:00:33.907
delta less than 0, so there are no real roots; the two roots are complex conjugates.

9
00:00:34.357 --> 00:00:38.885
Apply the quadratic formula with the square root of negative 1 equals i.

10
00:00:39.335 --> 00:00:40.229
Simplify.

11
00:00:41.029 --> 00:00:42.724
Collect the real solutions.

12
00:00:43.524 --> 00:00:47.077
Now look at the graph of the left side minus the right side.

13
00:00:47.527 --> 00:00:51.196
It crosses zero exactly at the solutions we found.

14
00:00:51.996 --> 00:01:01.389
So the answer is x equals negative 1; complex: 0.5 plus or minus 0.8660254i.

15
00:01:01.838 --> 00:01:14.029
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.

16
00:01:14.828 --> 00:01:21.898
x equals negative 1: left side equals 0, right side equals 0, which checks out.