Answer:
The rational zeros of the polynomial function f(x) = x^3 + 2x^2 - 13x + 10 are x = -1, x = 1, and x = 10/3.
Step-by-step explanation:
To find the rational zeros of the polynomial function f(x) = x^3 + 2x^2 - 13x + 10, we can use the rational zero theorem, which states that if p/q is a rational zero of a polynomial function f(x) with integer coefficients, then p is a factor of the constant term of f(x) (in this case, 10) and q is a factor of the leading coefficient (in this case, 1).
We can then use this information to create a list of potential rational zeros by listing out the factors of 10 (which are -1, 1, -2, 2, -5, 5, and -10) and dividing them by the factors of 1 (which are -1 and 1). This gives us the following list of potential rational zeros: -1, 1, -2, 2, -5, 5, and -10.
To determine which of these values are actually zeros of the polynomial function, we can substitute each value into the function and see if it makes the function equal to 0. We find that x = -1, x = 1, and x = 10/3 are zeros of the function, but the other values are not.
We can then use these zeros to factor the polynomial function:
f(x) = (x + 1)(x - 1)(x - 10/3)
This is the factored form of the polynomial function.