WEBVTT

1
00:00:00.000 --> 00:00:07.303
Here is the problem: 2x cubed plus 10x squared minus 4x minus 48 equals 0.

2
00:00:07.753 --> 00:00:11.015
A polynomial equation of degree 3 in x.

3
00:00:11.815 --> 00:00:13.324
Start with the equation.

4
00:00:13.774 --> 00:00:15.690
Move every term to one side.

5
00:00:16.140 --> 00:00:26.868
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
00:00:27.318 --> 00:00:38.255
Test candidates from the rational root theorem: x equals negative 3 is a root. Synthetic division by (x plus 3) leaves a polynomial of degree 1.

7
00:00:39.055 --> 00:00:48.912
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.

8
00:00:49.362 --> 00:00:51.115
Collect the real solutions.

9
00:00:51.915 --> 00:00:55.712
Now look at the graph of the left side minus the right side.

10
00:00:56.162 --> 00:00:59.889
It crosses zero exactly at the solutions we found.

11
00:01:00.689 --> 00:01:06.633
So the answer is x equals negative 4, x equals negative 3, x equals 2.

12
00:01:07.083 --> 00:01:18.925
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
00:01:19.725 --> 00:01:27.155
x equals negative 4: left side equals 0, right side equals 0, which checks out.

14
00:01:27.605 --> 00:01:35.372
x equals negative 3: left side equals 0, right side equals 0, which checks out.

15
00:01:35.822 --> 00:01:42.463
x equals 2: left side equals 0, right side equals 0, which checks out.