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To find all the zeros of \( f(x) = x^4 + 2x^3 - 13x^2 - 14x + 24 \), we can start by using the Rational Root Theorem to test for possible rational roots, and then apply synthetic division or polynomial division if necessary.

Let's go through the steps:

### Step 1: Apply the Rational Root Theorem
The Rational Root Theorem states that any rational root of the polynomial is a factor of the constant term divided by a factor of the leading coefficient. Here, the constant term is 24, and the leading coefficient is 1, so the possible rational roots are the factors of 24:
\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24.
\]

### Step 2: Test Possible Roots
We can test these values by substituting them into \( f(x) \) to see if any of them yield \( f(x) = 0 \).

### Step 3: Synthetic Division
If a value is a root, we can use synthetic division to factor it out, then repeat the process on the resulting polynomial until we get all roots.

Let me calculate these roots for you.The roots of the polynomial \( f(x) = x^4 + 2x^3 - 13x^2 - 14x + 24 \) are:

\[
x = -4, -2, 1, 3
\]

Since there are no repeated roots, we list each root only once. Arranged from smallest to largest, the zeros are:

\[
x = -4, -2, 1, 3
\]

Would you like further details on the steps, or do you have any questions?

Here are some related questions:
1. How would you use synthetic division to verify each root?
2. What is the Rational Root Theorem, and how does it help in finding polynomial roots?
3. What are imaginary roots, and when might they appear in polynomials?
4. How can you check for double roots or multiplicity in polynomials?
5. How does the degree of a polynomial relate to the maximum number of roots?

**Tip:** Checking possible rational roots first can save time, especially for higher-degree polynomials.

Find all the zeros of f(x) = x^4 + 2x^3 - 13x^2 - 14x + 24 and arrange the answers from smallest to largest. If there is a double root, list it twice.

Discover how to find all zeros of the polynomial function f(x) = x^4 + 2x^3 - 13x^2 - 14x + 24 using the Rational Root Theorem and synthetic division. This step-by-step solution is ideal for high school students studying polynomial functions and zeros.

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