Find all the zeros of the function below.
x^4+x^3-24x^2-25x-25
there has to be a number with an i
To find the zeros of the function, we can either factor the polynomial or use a numerical method such as synthetic division or the quadratic formula.
Factoring the polynomial:
We can try to factor the polynomial using synthetic division. Unfortunately, there are no rational roots. Therefore, we need to use another method.
Numerical Method:
Using a graphing calculator or computer software, we can plot the function and approximate the zeros. The graph shows that there are two real zeros and two complex zeros with the form a + bi.
The approximate real zeros are x ≈ -5 and x ≈ 1.758.
To find the complex zeros, we can use the other two roots and find the values of x when the polynomial equals 0.
Using x ≈ -5, we can perform synthetic division to find the quotient polynomial:
-5 | 1 1 -24 -25 -25
| -5 20 20 25
---------------------
1 -4 -4 -5 0
The quotient polynomial is x^3 - 4x^2 - 4x - 5. We can now find the rest of the zeros by factoring this polynomial.
Since there is no rational root, we can try synthetic division again using the real zero x ≈ 1.758:
1.758 | 1 -4 -4 -5
| 1.758 0.932 1.752
----------------------------
1 -2.242 -3.068 -3.248
The quotient is x^2 - 2.242x - 3.068. Factoring this quadratic equation, we get:
(x - 3.068)(x + 1) ≈ (x - 3.068)(x + 1) = 0
Therefore, the complex zeros are approximately x ≈ 3.068 and x ≈ -1.
The zeros of the function are -5, 1.758, 3.068 + i, and -1- i.