Question 1209690: Find all roots of the polynomial f(x) = 3x^3 - 7x^2 - 10x + 4 + 4x^3 - 5x^2 + 18x + 28.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 1. **Combine like terms:**
f(x) = (3x^3 + 4x^3) + (-7x^2 - 5x^2) + (-10x + 18x) + (4 + 28)
f(x) = 7x^3 - 12x^2 + 8x + 32
2. **Rational Root Theorem:** This theorem helps us find possible rational roots. Any rational root will be of the form p/q, where p is a factor of the constant term (32) and q is a factor of the leading coefficient (7).
Possible values for p: ±1, ±2, ±4, ±8, ±16, ±32
Possible values for q: ±1, ±7
Possible rational roots: ±1, ±2, ±4, ±8, ±16, ±32, ±1/7, ±2/7, ±4/7, ±8/7, ±16/7, ±32/7
3. **Test potential roots:** We can use synthetic division or direct substitution. Let's try x = -2:
```
-2 | 7 -12 8 32
| -14 52 -120
------------------
7 -26 60 -88
```
Since the remainder is not zero, -2 is not a root.
Let's try x = 4:
```
4 | 7 -12 8 32
| 28 64 288
------------------
7 16 72 320
```
Since the remainder is not zero, 4 is not a root.
4. **Continue testing or use a calculator:** Testing all the possible rational roots is tedious. It's often more efficient to use a graphing calculator or online polynomial solver to approximate the roots.
Using a calculator or solver, we find the roots are approximately:
* x ≈ -1.089
* x ≈ 1.402 + 1.494i
* x ≈ 1.402 - 1.494i
Notice that there is one real root and two complex roots. It's difficult to find these roots by hand. Numerical methods are the most practical approach.
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