What are the possible rational zeros of f(x) = 2x3 – 15x2 + 9x + 22?

any p/q where p divides 22 and q divides 2

To find the possible rational zeros of a polynomial, we can use the Rational Root Theorem. According to the theorem, if a rational number p/q is a zero of a polynomial with integer coefficients, then p is a factor of the constant term (22 in this case) and q is a factor of the leading coefficient (2 in this case).

In this case, the constant term is 22 and the leading coefficient is 2.

So, the possible rational zeros of f(x) = 2x^3 – 15x^2 + 9x + 22 come from the factors of 22 divided by the factors of 2.

The factors of 22 are: ±1, ±2, ±11, ±22
The factors of 2 are: ±1, ±2

Combining these sets of factors, the possible rational zeros are: ±1/1, ±1/2, ±2/1, ±2/2, ±11/1, ±11/2, ±22/1, ±22/2

Which simplifies to: ±1, ±1/2, ±2, ±11, ±11/2, ±22

So, the possible rational zeros of f(x) = 2x^3 – 15x^2 + 9x + 22 are: ±1, ±1/2, ±2, ±11, ±11/2, ±22.

To find the possible rational zeros of a polynomial function, you can use the Rational Root Theorem. The Rational Root Theorem states that if a polynomial has any rational zeros, they will be of the form p/q, where p is a factor of the constant term (in this case, 22) and q is a factor of the leading coefficient (in this case, 2).

For the polynomial function f(x) = 2x^3 - 15x^2 + 9x + 22, the factors of the constant term (22) are ±1, ±2, ±11, and ±22. The factors of the leading coefficient (2) are ±1 and ±2.

By substituting each possible value of p/q into the equation, you can determine which values, if any, are zeros of the function. In this case, you would evaluate f(p/q) = 2(p/q)^3 - 15(p/q)^2 + 9(p/q) + 22.

It is important to note that the Rational Root Theorem provides the possible rational zeros, but it does not guarantee that all of them will be zeros. You may need to use other techniques, such as synthetic division or factoring, to find the actual zeros of the function.