WEBVTT

1
00:00:00.000 --> 00:00:06.142
Here is the problem: x to the power 5 minus 5x cubed plus 4x equals 0.

2
00:00:06.592 --> 00:00:09.889
A polynomial equation of degree 5 in x.

3
00:00:10.689 --> 00:00:12.187
Start with the equation.

4
00:00:12.637 --> 00:00:14.599
Move every term to one side.

5
00:00:15.049 --> 00:00:25.869
Test candidates from the rational root theorem: x equals 0 is a root. Synthetic division by (x minus 0) leaves a polynomial of degree 4.

6
00:00:26.319 --> 00:00:36.873
Test candidates from the rational root theorem: x equals negative 2 is a root. Synthetic division by (x plus 2) leaves a polynomial of degree 3.

7
00:00:37.672 --> 00:00:47.482
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.

8
00:00:47.932 --> 00:00:58.218
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.

9
00:00:58.668 --> 00:01:08.629
Test candidates from the rational root theorem: x equals 2 is a root. Synthetic division by (x minus 2) leaves a polynomial of degree 0.

10
00:01:09.079 --> 00:01:10.774
Collect the real solutions.

11
00:01:11.574 --> 00:01:20.189
So the answer is x equals negative 2, x equals negative 1, x equals 0, x equals 1, x equals 2.

12
00:01:20.639 --> 00:01:32.632
A degree-5 polynomial has exactly 5 roots counted with multiplicity (some may be complex). Exact rational roots are found first, then the rest are solved exactly or approximated.

13
00:01:33.432 --> 00:01:40.479
x equals negative 2: left side equals 0, right side equals 0, which checks out.

14
00:01:40.929 --> 00:01:48.057
x equals negative 1: left side equals 0, right side equals 0, which checks out.

15
00:01:48.508 --> 00:01:55.137
x equals 0: left side equals 0, right side equals 0, which checks out.

16
00:01:55.587 --> 00:02:02.123
x equals 1: left side equals 0, right side equals 0, which checks out.