Find all the zeros of the function below.
x^4+x^3-24x^2-25x-25
To find the zeros of the function, we need to find the values of x for which the function equals zero.
We can try factorizing the polynomial, but it does not seem to factorize easily.
Alternatively, we can use the Rational Root Theorem to check for possible rational zeros.
The Rational Root Theorem states that if a polynomial equation has a rational root p/q (in simplest form), then p must be a factor of the constant term (in this case, -25) and q must be a factor of the leading coefficient (in this case, 1).
The possible rational zeros of the function are therefore given by: ±1, ±5, ±25.
To check if these values are zeros of the function, we substitute them into the equation:
For x = -1: (-1)^4 + (-1)^3 - 24(-1)^2 - 25(-1) - 25 = 1 - 1 + 24 + 25 - 25 = 24
For x = 1: (1)^4 + (1)^3 - 24(1)^2 - 25(1) - 25 = 1 + 1 - 24 - 25 - 25 = -72
For x = -5: (-5)^4 + (-5)^3 - 24(-5)^2 - 25(-5) - 25 = 625 - 125 + 600 + 125 - 25 = 1150
For x = 5: (5)^4 + (5)^3 - 24(5)^2 - 25(5) - 25 = 625 + 125 - 600 - 125 - 25 = 0
For x = -25: (-25)^4 + (-25)^3 - 24(-25)^2 - 25(-25) - 25 = 390625 - 15625 + 150000 - 625 - 25 = 525000
For x = 25: (25)^4 + (25)^3 - 24(25)^2 - 25(25) - 25 = 390625 + 15625 - 150000 - 625 - 25 = 24125
Therefore, the zeros of the function are x = 5 and x = -25.