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 = $\frac{9}{2}$ , x = $1$, x = $-8$
f(x) = __________ x^3 + __________ x^2 + __________ x + __________
1. **State the problem:** We need to find the simplest polynomial function with integer coefficients and positive leading coefficient that has zeros at $x=\frac{9}{2}$, $x=1$, and $x=-8$.
2. **Recall the formula:** If a polynomial has zeros $r_1$, $r_2$, and $r_3$, then it 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:** Here,
$$f(x) = a\left(x - \frac{9}{2}\right)(x - 1)(x + 8)$$
4. **Clear the fraction to get integer coefficients:** Multiply the factor with fraction by 2 inside to avoid fractions:
$$f(x) = a\left(2x - 9\right)(x - 1)(x + 8)$$
We choose $a=1$ to keep the leading coefficient positive and simplest.
5. **Expand step-by-step:**
First expand $(x - 1)(x + 8)$:
$$ (x - 1)(x + 8) = x^2 + 8x - x - 8 = x^2 + 7x - 8 $$
6. **Multiply $(2x - 9)$ by the quadratic:**
$$ (2x - 9)(x^2 + 7x - 8) = 2x(x^2 + 7x - 8) - 9(x^2 + 7x - 8) $$
$$ = 2x^3 + 14x^2 - 16x - 9x^2 - 63x + 72 $$
7. **Combine like terms:**
$$ 2x^3 + (14x^2 - 9x^2) + (-16x - 63x) + 72 = 2x^3 + 5x^2 - 79x + 72 $$
8. **Final polynomial:**
$$ f(x) = 2x^3 + 5x^2 - 79x + 72 $$
This polynomial has integer coefficients, leading coefficient positive, and zeros at $\frac{9}{2}$, $1$, and $-8$.
**Answer:**
$f(x) = 2x^3 + 5x^2 - 79x + 72$