WEBVTT

1
00:00:00.000 --> 00:00:07.128
Here is the problem: x cubed plus 2x squared minus 16x minus 32 equals 0.

2
00:00:07.579 --> 00:00:10.772
A polynomial equation of degree 3 in x.

3
00:00:11.572 --> 00:00:13.058
Start with the equation.

4
00:00:13.508 --> 00:00:15.447
Move every term to one side.

5
00:00:15.897 --> 00:00:26.172
Test candidates from the rational root theorem: x equals negative 4 is a root. Synthetic division by (x plus 4) leaves a polynomial of degree 2.

6
00:00:26.622 --> 00:00:37.257
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 1.

7
00:00:38.057 --> 00:00:48.042
Test candidates from the rational root theorem: x equals 4 is a root. Synthetic division by (x minus 4) leaves a polynomial of degree 0.

8
00:00:48.492 --> 00:00:50.199
Collect the real solutions.

9
00:00:50.999 --> 00:00:54.946
Now look at the graph of the left side minus the right side.

10
00:00:55.396 --> 00:00:59.146
It crosses zero exactly at the solutions we found.

11
00:00:59.946 --> 00:01:06.157
So the answer is x equals negative 4, x equals negative 2, x equals 4.

12
00:01:06.607 --> 00:01:18.542
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
00:01:19.342 --> 00:01:26.563
x equals negative 4: left side equals 0, right side equals 0, which checks out.

14
00:01:27.013 --> 00:01:34.107
x equals negative 2: left side equals 0, right side equals 0, which checks out.

15
00:01:34.557 --> 00:01:41.488
x equals 4: left side equals 0, right side equals 0, which checks out.