The given polynomial is f(x) = x³ - 3x² + 4 = 0
x=2 is a zero with a multiplicity of 2, therefore (x - 2)² should be a factor of f(x).
Perform synthetic division with (x-2).
2 | 1 -3 0 4
2 -2 -4
-------------------
1 -1 -2 0
Therefore f(x) = (x-2)(x² - x - 2)
Perform another synthetic division on x² - x - 2 (or factorize).
2 | 1 -1 -2
2 2
---------------
1 1 0
Therefore f(x) = (x-2)(x-2)(x+1) = (x-2)²(x+1)
Note:
If you do not know synthetic division, use long division.
For example,
x² - x - 2
---------------------------------
x-2 | x³ - 3x² + 0x + 4
x³ - 2x²
-------------------------------
- x² + 0x + 4
- x² + 2x
----------------------
- 2x + 4
- 2x + 4
Answer: f(x) = (x+1)(x-2)²