how to factorize x^3-18x^2+31X-o=0
To factorize the given expression, x^3 - 18x^2 + 31x - o = 0, we can use the Rational Root Theorem and synthetic division. Here's how:
Step 1: Apply the Rational Root Theorem. This theorem states that if a polynomial has a rational root, it will be of the form p/q, where p is a factor of the constant term (in this case, 0), and q is a factor of the leading coefficient (in this case, 1).
- Factors of the constant term: ±1, ±o
- Factors of the leading coefficient: ±1
Step 2: Test the potential rational roots using synthetic division to find the actual roots. Start with p/q as the divisor and see if the remainder (after dividing) is zero.
- Start with p = ±1:
- When p = 1, synthetic division gives us remainder ≠ 0.
- When p = -1, synthetic division gives us remainder = 0.
- Therefore, (x + 1) is a factor.
Step 3: Perform synthetic division using (x + 1) as the divisor to obtain the reduced quadratic equation.
- Synthetic division steps:
- -1 | 1 -18 31 0
| -1 19 -50
| 1 -19 50 -50
- The reduced quadratic equation is: x^2 - 19x + 50 = 0
Step 4: Factorize the reduced quadratic equation.
- For x^2 - 19x + 50 = 0:
- You can either use factoring, completing the square, or the quadratic formula.
- By factoring, we have: (x - 5)(x - 10) = 0.
Step 5: Combine the factors.
- The quadratic factors of the reduced equation: (x - 5)(x - 10)
- The linear factor from earlier: (x + 1)
- Combining them, the solution is: (x + 1)(x - 5)(x - 10) = 0
Therefore, the factored form of the given equation is (x + 1)(x - 5)(x - 10) = 0.