Let's break down each part of the question.

Part 1a: Find all zeros of the function $f(x) = -x^3 + 4x^2 + 15x - 18$

To find the zeros, we need to solve the equation $f(x) = 0$:

$-x^3 + 4x^2 + 15x - 18 = 0$

We can attempt to factor this polynomial or apply the Rational Root Theorem to test possible rational roots. Commonly, integer factors of the constant term (-18) divided by factors of the leading coefficient (-1) yield possible rational roots: $\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$.

Testing these values, we find that $x = 3$ is a root (since substituting it into the equation gives zero). We can then factor out $(x - 3)$ from the polynomial by performing synthetic division:

1. Divide $-x^3 + 4x^2 + 15x - 18$ by $(x - 3)$.
2. The result is $-(x - 3)(x^2 + x + 6)$.
3. Solving $x^2 + x + 6 = 0$ gives complex roots (using the quadratic formula).

The zeros are: $x = 3, x = -\frac{1}{2} + \frac{\sqrt{23}}{2}i, x = -\frac{1}{2} - \frac{\sqrt{23}}{2}i$

Part 1b: Draw a complete graph of the function without a graphing utility

Given the polynomial $f(x) = -x^3 + 4x^2 + 15x - 18$, observe the following:

• The function has a negative leading coefficient and an odd degree, meaning it will fall to the right and rise to the left.
• The real root $x = 3$ means the graph crosses the x-axis at this point.
• The function’s end behavior and root at $x = 3$ guide the general shape: a curve in the second quadrant falling toward a minimum in the third quadrant, then rising to cross the x-axis at $x = 3$.

This suggests graph choice B is correct based on the described behavior and the placement of the curve through the origin and near $x = 3$.

Would you like further details or clarification on this process?

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1. How do you use synthetic division to factor polynomials?
2. What is the Rational Root Theorem, and how does it help find polynomial roots?
3. How can we determine the end behavior of polynomial functions?
4. Why do complex roots always appear in conjugate pairs?
5. How can the quadratic formula help find complex roots of a polynomial?

Tip: When factoring higher-degree polynomials, check for any obvious real roots first using the Rational Root Theorem to simplify the problem.