what are all the real zeros for 1x^5+2x^4+4x^3+19x^2+9x-5?
To find the real zeros of the polynomial 1x^5 + 2x^4 + 4x^3 + 19x^2 + 9x - 5, we can use a method called the Rational Root Theorem and synthetic division.
1. The Rational Root Theorem states that any rational root (which can be expressed as a fraction) of the polynomial must be a factor of the constant term (in this case, -5) divided by a factor of the leading coefficient (in this case, 1). So the potential rational roots are ±1, ±5.
2. We can use synthetic division to check if any of these potential roots are actual zeros of the equation.
- First, we test x = -1:
| -1 | 1 2 4 19 9 -5 |
-1 -1 -3 -16 7 -2
This means that x + 1 is not a factor, as the remainder is not zero.
- Next, we test x = 1:
| 1 | 1 2 4 19 9 -5 |
1 3 7 26 35 30
Again, x - 1 is not a factor since the remainder is not zero.
- Continuing this process for x = ±5, we find that none of these potential roots are actual zeros.
3. Therefore, we cannot find any real zeros using the Rational Root Theorem alone. We may need to use numerical methods or advanced techniques like the Descartes' Rule of Signs, the Complex Conjugate Root Theorem, or the graphing calculator to find the real zeros of this polynomial.
Please note that this is just a method to find potential real zeros. The actual process of finding the real zeros might involve more advanced techniques depending on the nature of the polynomial equation.