To solve the cubic equation \(x^3 - 78x^2 + 2028x - 17576 = 0\), let's try factoring by grouping or checking for possible rational roots using the Rational Root Theorem.

### Step 1: Use the Rational Root Theorem
The Rational Root Theorem suggests that any rational root of the polynomial is of the form \( \frac{p}{q} \), where \( p \) is a divisor of the constant term (here \(-17576\)) and \( q \) is a divisor of the leading coefficient (here \(1\)).

The divisors of \(-17576\) are possible candidates for roots. We can try simple values to find a possible root.

### Step 2: Trial and error (Checking potential roots)
Let's try \( x = 26 \):

\[
26^3 - 78 \times 26^2 + 2028 \times 26 - 17576 = 0
\]

Simplifying:

\[
17576 - 52728 + 52728 - 17576 = 0
\]

Since \( x = 26 \) is a root, we can factor \( (x - 26) \) from the cubic polynomial.

### Step 3: Factor the cubic polynomial
Now that we know \( x = 26 \) is a root, we can divide the polynomial by \( (x - 26) \) to get the remaining quadratic.

Dividing \( x^3 - 78x^2 + 2028x - 17576 \) by \( x - 26 \), we get:

\[
x^3 - 78x^2 + 2028x - 17576 = (x - 26)(x^2 - 52x + 676)
\]

### Step 4: Solve the quadratic equation
Now solve \( x^2 - 52x + 676 = 0 \).

Use the quadratic formula:

\[
x = \frac{-(-52) \pm \sqrt{(-52)^2 - 4(1)(676)}}{2(1)}
\]

Simplifying:

\[
x = \frac{52 \pm \sqrt{2704 - 2704}}{2}
\]

\[
x = \frac{52 \pm \sqrt{0}}{2}
\]

\[
x = \frac{52}{2} = 26
\]

Thus, the quadratic has a double root at \( x = 26 \).

### Final Answer:
The solution to the equation \(x^3 - 78x^2 + 2028x - 17576 = 0\) is:

\[
x = 26 \quad \text{(triple root)}
\]

Would you like more details or have any questions?

Here are five related questions to further your understanding:

1. How do you use the Rational Root Theorem to find possible rational roots?
2. How do you divide polynomials using synthetic division?
3. What is the significance of a root appearing more than once?
4. Can cubic equations have more than one distinct real root?
5. How can the quadratic formula be applied to non-factorable quadratics?

**Tip:** For cubic equations, always check for rational roots first, as it simplifies the solving process if a root can be easily factored out.

Solve the cubic equation x^3 - 78x^2 + 2028x - 17576 = 0 using the Rational Root Theorem and quadratic formula. This step-by-step solution covers key algebraic concepts, including factoring cubic polynomials and solving quadratic equations. Ideal for students in Grades 10-12.

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