How to factor x^3 - 3x^2 + 4 =0
Use D'Alembert's Rational Roots Theorem. Any rational roots of the form of p/q (p and q assumed to be relatively prime) must be such that p divides the constant term (in this case 4) and q divides the coefficient of the largest power of x (in this case 1).
So, the candidate roots are ±1, ±2, and ±4. If you try these, you see that
x = -1 and x = 2 are roots. There are no other roots, but there must be three roots when we count by multiplicity. So, one of the two roots we found must have a multiplicity of 2.
It isn't difficult to see that this must be the root x = 2, because then you get the correct factorization:
(x+1)(x-2)^2
To factor the cubic polynomial x^3 - 3x^2 + 4 = 0, we can use the rational roots theorem.
1. First, we need to determine the possible rational roots of the polynomial. According to the rational roots theorem, the numerator of the rational root should divide the constant term (4 in this case), and the denominator should divide the coefficient of the highest degree term (1 in this case).
2. The possible rational roots can be obtained by listing all the factors of the constant term (4) and dividing them by the factors of the coefficient of the highest degree term (1). In this case, the factors of 4 are ±1, ±2, and ±4, and the factors of 1 are ±1.
3. We now have a list of possible rational roots: ±1, ±2, and ±4.
4. To determine which roots are actually solutions, we can substitute each of the possible rational roots into the polynomial and check if the result is zero.
- If a rational root produces a zero result, then it is a solution.
- If a rational root does not produce a zero result, then it is not a solution.
In this case, you can try substituting each possible root into the polynomial:
For x = -1: (-1)^3 - 3(-1)^2 + 4 = -1 - 3 + 4 = 0 (The polynomial evaluates to zero, so -1 is a solution.)
For x = 2: (2)^3 - 3(2)^2 + 4 = 8 - 12 + 4 = 0 (The polynomial evaluates to zero, so 2 is a solution.)
The other possible roots do not give a zero result.
5. We have found two solutions: x = -1 and x = 2.
6. Since we have a cubic polynomial, there should be three roots in total. Therefore, one of the solutions (in this case, x = 2) must have a multiplicity of 2.
7. We can represent the factorization of the polynomial as (x + 1)(x - 2)^2. The term (x - 2)^2 signifies that the root x = 2 has a multiplicity of 2.
Therefore, the factored form of the polynomial x^3 - 3x^2 + 4 = 0 is (x + 1)(x - 2)^2.