WEBVTT

1
00:00:00.000 --> 00:00:03.889
Here is the problem: (x minus 1) cubed equals 0.

2
00:00:04.339 --> 00:00:07.532
A polynomial equation of degree 3 in x.

3
00:00:08.332 --> 00:00:09.818
Start with the equation.

4
00:00:10.268 --> 00:00:12.137
Move every term to one side.

5
00:00:12.587 --> 00:00:23.071
Test candidates from the rational root theorem: x equals 1 is a root. Synthetic division by (x minus 1) leaves a polynomial of degree 2.

6
00:00:23.521 --> 00:00:33.715
Test candidates from the rational root theorem: x equals 1 is a root. Synthetic division by (x minus 1) leaves a polynomial of degree 1.

7
00:00:34.515 --> 00:00:44.221
Test candidates from the rational root theorem: x equals 1 is a root. Synthetic division by (x minus 1) leaves a polynomial of degree 0.

8
00:00:44.671 --> 00:00:46.447
Collect the real solutions.

9
00:00:47.247 --> 00:00:50.881
Now look at the graph of the left side minus the right side.

10
00:00:51.331 --> 00:00:55.046
It crosses zero exactly at the solutions we found.

11
00:00:55.846 --> 00:00:57.971
So the answer is x equals 1.

12
00:00:58.421 --> 00:01:10.356
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
00:01:11.156 --> 00:01:17.843
x equals 1: left side equals 0, right side equals 0, which checks out.