Question 1209689: Find all roots of the polynomial f(x) = 2x^3 - 5x^2 - 2x + 2 - 8x^3 + 11x^2 - 17x + 8.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 1. **Combine like terms:**
f(x) = (2x^3 - 8x^3) + (-5x^2 + 11x^2) + (-2x - 17x) + (2 + 8)
f(x) = -6x^3 + 6x^2 - 19x + 10
2. **Rational Root Theorem:** Possible rational roots are of the form p/q, where p is a factor of 10 and q is a factor of -6.
Possible values for p: ±1, ±2, ±5, ±10
Possible values for q: ±1, ±2, ±3, ±6
Possible rational roots: ±1, ±2, ±5, ±10, ±1/2, ±5/2, ±1/3, ±2/3, ±5/3, ±10/3, ±1/6, ±5/6
3. **Test potential roots:** This is tedious. Let's use a graphing calculator or online polynomial solver.
Using a calculator, we find the roots are approximately:
* x ≈ 0.215 + 1.696i
* x ≈ 0.215 - 1.696i
* x ≈ 0.570
There is one real root and two complex roots. It is very difficult to find these roots by hand. Numerical methods are the most practical approach.
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