To find the zeros of the polynomial function $f(x) = x^3 + 7x^2 - 4x - 28$, we need to solve for $x$ such that $f(x) = 0$. Let's proceed step by step:

Step 1: Use the Rational Root Theorem

The possible rational roots are the factors of the constant term (-28) divided by the factors of the leading coefficient (1). These are:

$\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28$

Step 2: Test Possible Roots

We'll substitute these values into the function to see if they yield $f(x) = 0$.

Test $x = -4$:

$f(-4) = (-4)^3 + 7(-4)^2 - 4(-4) - 28$ $= -64 + 112 + 16 - 28 = 0$

So, $x = -4$ is a root.

Step 3: Synthetic Division

Now, divide $f(x)$ by $(x + 4)$ using synthetic or polynomial division.

$f(x) = (x + 4)(x^2 + 3x - 7)$

Step 4: Solve the Quadratic Equation

Now solve $x^2 + 3x - 7 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = 3$, $c = -7$: $x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-7)}}{2(1)}$ $x = \frac{-3 \pm \sqrt{9 + 28}}{2}$ $x = \frac{-3 \pm \sqrt{37}}{2}$

Final Answer

The zeros of the polynomial are: $x = -4, \quad x = \frac{-3 + \sqrt{37}}{2}, \quad x = \frac{-3 - \sqrt{37}}{2}$

Would you like a detailed explanation of any step?

Related Questions:

1. How do you perform synthetic division for polynomials?
2. What is the Rational Root Theorem and how does it work?
3. How do you solve quadratic equations using the quadratic formula?
4. Can you explain why the zeros of a polynomial are also called roots?
5. How would this process change if the polynomial degree were higher than 3?

Tip: When testing possible rational roots, start with smaller integers, as they are easier to calculate and often reveal at least one root quickly.