WEBVTT

1
00:00:00.000 --> 00:00:05.758
Here is the problem: x cubed minus 3x squared minus 4x plus 12.

2
00:00:06.208 --> 00:00:09.041
Factor a polynomial over the rational numbers.

3
00:00:09.841 --> 00:00:12.291
Write the polynomial in standard form.

4
00:00:12.741 --> 00:00:27.137
Apply the Rational Root Theorem. Any rational root p/q has p dividing the constant term 12 and q dividing the leading coefficient 1. Testing these candidates gives the roots negative 2, 2, 3.

5
00:00:27.587 --> 00:00:32.254
Divide by (x plus 2) using synthetic division at x equals negative 2.

6
00:00:32.704 --> 00:00:37.371
Divide by (x minus 2) using synthetic division at x equals 2.

7
00:00:38.171 --> 00:00:43.210
Divide by (x minus 3) using synthetic division at x equals 3.

8
00:00:43.660 --> 00:00:45.483
Write the complete factorization.

9
00:00:46.283 --> 00:00:50.637
So the answer is (x minus 3)(x minus 2)(x plus 2).

10
00:00:51.087 --> 00:01:00.654
x cubed minus 3x squared minus 4x plus 12 factors completely over the rationals as (x minus 3)(x minus 2)(x plus 2).

11
00:01:01.454 --> 00:01:10.103
Expanding (x minus 3)(x minus 2)(x plus 2) gives back x cubed minus 3x squared minus 4x plus 12.