Question: Write the simplest polynomial function using the three zeros given, where all coefficients are integers and the leading coefficient is positive. Enter your answers below.
zeros at x = 1, x = - \frac{1}{2}, x = 8
f(x) =
x^3 + x^2 + x +
1. **State the problem:** Find the simplest polynomial function with integer coefficients and positive leading coefficient given zeros at $x=1$, $x=-\frac{1}{2}$, and $x=8$.
2. **Recall the formula:** If a polynomial has zeros $r_1$, $r_2$, $r_3$, then the polynomial can be written as
$$f(x) = a(x-r_1)(x-r_2)(x-r_3)$$
where $a$ is the leading coefficient.
3. **Apply the zeros:**
$$f(x) = a\left(x-1\right)\left(x - \left(-\frac{1}{2}\right)\right)(x-8) = a(x-1)\left(x + \frac{1}{2}\right)(x-8)$$
4. **Clear the fraction:** To have integer coefficients, multiply the factor with fraction by 2 inside the polynomial by choosing $a=2$:
$$f(x) = 2(x-1)\left(x + \frac{1}{2}\right)(x-8) = (x-1)(2x+1)(x-8)$$
5. **Expand step-by-step:**
First expand $(x-1)(2x+1)$:
$$ (x-1)(2x+1) = 2x^2 + x - 2x - 1 = 2x^2 - x - 1 $$
6. **Multiply by $(x-8)$:**
$$ f(x) = (2x^2 - x - 1)(x-8) $$
$$ = 2x^3 - 16x^2 - x^2 + 8x - x + 8 $$
$$ = 2x^3 - 17x^2 + 7x + 8 $$
7. **Final polynomial:**
$$ \boxed{f(x) = 2x^3 - 17x^2 + 7x + 8} $$
This polynomial has integer coefficients, positive leading coefficient, and the given zeros.