Answer:
2 r^2 + 18 r + 27 (2 r - 3)
Explanation:
Factor the following:
4 r^3 + 30 r^2 - 81
The possible rational roots of 4 r^3 + 30 r^2 - 81 are r = ± 1/4, r = ± 3/4, r = ± 9/4, r = ± 27/4, r = ± 81/4, r = ± 1/2, r = ± 3/2, r = ± 9/2, r = ± 27/2, r = ± 81/2, r = ± 1, r = ± 3, r = ± 9, r = ± 27, r = ± 81. Of these, r = 3/2 is a root. This gives 2 r - 3 as all linear factors:
((2 r - 3) (4 r^3 + 30 r^2 - 81))/(2 r - 3)
| |
2 r | - | 3 | | 2 r^2 | + | 18 r | + | 27
4 r^3 | + | 30 r^2 | + | 0 | - | 81
4 r^3 | - | 6 r^2 | | | |
| | 36 r^2 | + | 0 | |
| | 36 r^2 | - | 54 r | |
| | | | 54 r | - | 81
| | | | 54 r | - | 81
| | | | | | 0:
Answer: 2 r^2 + 18 r + 27 (2 r - 3)