Let's solve this step-by-step.

Problem Analysis

1. The given polynomial function is: $f(x) = x^3 - 7x^2 - 41x - 33$
2. We're asked to find all real zeros of $f(x)$ using the Rational Zeros Theorem.
3. Then, use these zeros to factor $f(x)$.

Step 1: Applying the Rational Zeros Theorem

The Rational Zeros Theorem suggests that any rational solution for a polynomial equation with integer coefficients can be expressed as $\pm \frac{p}{q}$, where:

• $p$ is a factor of the constant term, here $-33$.
• $q$ is a factor of the leading coefficient, here $1$.

Factors of $-33$: $\pm 1, \pm 3, \pm 11, \pm 33$

Factors of $1$: $\pm 1$

So, the possible rational roots are $\pm 1, \pm 3, \pm 11, \pm 33$.

Step 2: Testing Possible Zeros

Using synthetic division or direct substitution, we find that:

• $x = -1$, $x = 11$, and $x = -3$ are zeros of $f(x)$.

Step 3: Writing $f(x)$ in Factored Form

Since the real zeros are $x = -1$, $x = 11$, and $x = -3$, we can express $f(x)$ as: $f(x) = (x + 1)(x - 11)(x + 3)$

Final Answer

The factored form of $f(x)$ is: $f(x) = (x + 1)(x - 11)(x + 3)$

Would you like more details on any step, or have additional questions?

Follow-up Questions

1. How does the Rational Zeros Theorem help in identifying possible roots?
2. What are alternative methods for finding real zeros in polynomials?
3. Can synthetic division verify if a candidate zero is correct?
4. How can we expand a factored polynomial back into standard form?
5. What are the benefits of factoring a polynomial?

Tip:

When using the Rational Zeros Theorem, always check possible rational roots systematically with synthetic division or substitution to confirm zeros efficiently.