WEBVTT

1
00:00:00.000 --> 00:00:03.274
Here is the problem: x cubed plus 27.

2
00:00:03.724 --> 00:00:06.441
Factor a polynomial over the rational numbers.

3
00:00:07.241 --> 00:00:09.528
Write the polynomial in standard form.

4
00:00:09.978 --> 00:00:19.533
Recognise a special pattern. The remaining factor is a sum of cubes: a cubed plus b cubed equals (a plus b)(a squared minus ab plus b squared).

5
00:00:19.983 --> 00:00:32.870
Apply the Rational Root Theorem. Any rational root p/q has p dividing the constant term 27 and q dividing the leading coefficient 1. Testing these candidates gives the root negative 3.

6
00:00:33.320 --> 00:00:38.138
Divide by (x plus 3) using synthetic division at x equals negative 3.

7
00:00:38.938 --> 00:00:46.426
Check the remaining quadratic. Its discriminant is negative 27, negative, so it is irreducible over the rationals.

8
00:00:46.876 --> 00:00:48.641
Write the complete factorization.

9
00:00:49.441 --> 00:00:54.108
So the answer is (x plus 3)(x squared minus 3x plus 9).

10
00:00:54.558 --> 00:01:02.708
x cubed plus 27 factors completely over the rationals as (x plus 3)(x squared minus 3x plus 9).

11
00:01:03.508 --> 00:01:10.683
Expanding (x plus 3)(x squared minus 3x plus 9) gives back x cubed plus 27.