Answer:
(a, b, c) = (2, -6, 2) or (2, 2, -6)
Explanation:
Expand the expression on the right and match coefficients, or factor the expression on the left and match factors.
2x³ -8x² -24x = ax(x +b)(x +c)
2x(x² -4x -12) = ax(x +b)(x +c)
2x(x -6)(x +2) = ax(x +b)(x +c)
In this form, we can match the numbers to get ...
You may recognize that b and c can be interchanged.
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Alternate solution
If you expand the expression on the right, you get ...
= ax³ +a(b+c)x² +abcx
This gives you 3 equations:
- a = 2
- 2(b +c) = -8
- 2bc = -24
This is basically the same factoring problem in another form. You have ...
bc = -12
b+c = -4
The factors of -12 are ...
-12 = 1(-12) = 2(-6) = 3(-4) . . . . . . with sums -11, -4, -1
So, b and c are -6 and 2, in no particular order.
(a, b, c) = (2, -6, 2) or (2, 2, -6)