WEBVTT

1
00:00:00.000 --> 00:00:03.866
Here is the problem: x to the power 4 equals 16.

2
00:00:04.316 --> 00:00:07.532
A polynomial equation of degree 4 in x.

3
00:00:08.332 --> 00:00:09.806
Start with the equation.

4
00:00:10.256 --> 00:00:12.195
Move every term to one side.

5
00:00:12.645 --> 00:00:23.454
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.

6
00:00:23.904 --> 00:00:33.970
Test candidates from the rational root theorem: x equals 2 is a root. Synthetic division by (x minus 2) leaves a polynomial of degree 2.

7
00:00:34.770 --> 00:00:36.407
Solve the remaining quadratic.

8
00:00:36.857 --> 00:00:38.262
Compute the discriminant.

9
00:00:38.712 --> 00:00:44.726
delta less than 0, so there are no real roots; the two roots are complex conjugates.

10
00:00:45.176 --> 00:00:49.774
Apply the quadratic formula with the square root of negative 1 equals i.

11
00:00:50.574 --> 00:00:51.491
Simplify.

12
00:00:51.941 --> 00:00:53.694
Collect the real solutions.

13
00:00:54.494 --> 00:00:58.174
Now look at the graph of the left side minus the right side.

14
00:00:58.624 --> 00:01:02.258
It crosses zero exactly at the solutions we found.

15
00:01:03.058 --> 00:01:10.186
So the answer is x equals negative 2, x equals 2; complex: plus or minus 2i.

16
00:01:10.637 --> 00:01:22.549
A degree-4 polynomial has exactly 4 roots counted with multiplicity (some may be complex). Exact rational roots are found first, then the rest are solved exactly or approximated.

17
00:01:23.349 --> 00:01:30.953
x equals negative 2: left side equals 16, right side equals 16, which checks out.

18
00:01:31.403 --> 00:01:38.915
x equals 2: left side equals 16, right side equals 16, which checks out.