To find the possible rational zeros of the polynomial function f(x) = 2x³ + 15x² + 9x + 22, we can use the Rational Root Theorem. This theorem states that any rational solution, expressed as a fraction p/q, has p as a factor of the constant term and q as a factor of the leading coefficient.
In this polynomial, the constant term is 22 and the leading coefficient is 2.
The factors of 22 are: ±1, ±2, ±11, ±22.
The factors of 2 are: ±1, ±2.
Now, we can form the possible rational zeros by taking each factor of the constant term and dividing it by each factor of the leading coefficient:
- From ±1: ±1/1 = ±1
- From ±1: ±1/2 = ±1/2
- From ±2: ±2/1 = ±2
- From ±2: ±2/2 = ±1
- From ±11: ±11/1 = ±11
- From ±11: ±11/2 = ±11/2
- From ±22: ±22/1 = ±22
- From ±22: ±22/2 = ±11
After simplifying, we combine all possible values. The final list includes:
- ±1
- ±1/2
- ±2
- ±11
- ±11/2
- ±22
Thus, the possible rational zeros of the polynomial function f(x) = 2x³ + 15x² + 9x + 22 are: ±1, ±1/2, ±2, ±11, ±11/2, ±22.