what are the rational zeros of x^3 - x^2 - 10x -8 = 0
first guess one zero, like -1
try that -1
-1 - 1 -10(-1) - 8
-2 + 10 -8
0
ok, x = -1 works
so (x+1) is a factor
divide by long division or synthetic division
(x^3 - x^2 - 10x -8)/(x+1)
I get
x^2 - 2 x - 8
so
(x+1)(x^2 - 2 x - 8 ) = 0
get your next two roots by solving the quadratic
x^2 - 2 x - 8 = 0
I can not factor that so use the quadratic equation.
you can factor x^2 - 2x - 8 = 0
(x-4)(x+2)=0
therefore the three zeros [or 'roots' as I know them as] are x = 4, -1, -2
To find the rational zeros of a polynomial equation, we can use the Rational Root Theorem. According to the theorem, if a rational number p/q is a root of a polynomial equation, then p is a factor of the constant term (in this case -8), and q is a factor of the leading coefficient (in this case 1).
For the given equation x^3 - x^2 - 10x - 8 = 0, the leading coefficient is 1 and the constant term is -8. Therefore, all the possible rational zeros can be obtained by taking all the factors of -8 and dividing them by all the factors of 1.
The factors of -8 are: -1, -2, -4, 8, 1, 2, 4, -8
The factors of 1 are: -1, 1
So, the possible rational zeros of the equation can be: -1, -2, -4, 8, 1, 2, 4, or -8.
To find out which of these values are actual zeros of the equation, we can use a method such as synthetic division or substitution. By trying out each of these possible zeros, we can determine which ones satisfy the equation.
Note: It is important to remember that these are the possible rational zeros, but there is no guarantee that any of them will be an actual zero.