To solve the cubic equation \( a^3 - 26a + 70 = 0 \), we can attempt factoring or use numerical methods like the Rational Root Theorem to find possible rational roots.

### Step 1: Apply the Rational Root Theorem
The Rational Root Theorem suggests trying factors of the constant term (70) over factors of the leading coefficient (1). The possible rational roots are the factors of 70, which are:

\[
\pm 1, \pm 2, \pm 5, \pm 7, \pm 10, \pm 14, \pm 35, \pm 70
\]

### Step 2: Test for possible roots
We start by testing these values to see if they satisfy the equation \( a^3 - 26a + 70 = 0 \).

- For \( a = 2 \):
  \[
  2^3 - 26(2) + 70 = 8 - 52 + 70 = 26 \neq 0
  \]

- For \( a = 5 \):
  \[
  5^3 - 26(5) + 70 = 125 - 130 + 70 = 65 \neq 0
  \]

- For \( a = -5 \):
  \[
  (-5)^3 - 26(-5) + 70 = -125 + 130 + 70 = 75 \neq 0
  \]

- For \( a = 7 \):
  \[
  7^3 - 26(7) + 70 = 343 - 182 + 70 = 231 \neq 0
  \]

- For \( a = -7 \):
  \[
  (-7)^3 - 26(-7) + 70 = -343 + 182 + 70 = -91 \neq 0
  \]

None of these work. However, further exploration using numerical methods or factoring could help find the exact solutions. Alternatively, graphical or iterative methods might yield approximate solutions.

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**Relative Questions**:
1. How would you solve cubic equations with irrational roots?
2. What is the process for using synthetic division in cubic equations?
3. Can numerical methods like Newton’s method be applied to this equation?
4. How do graphical solutions for cubic equations work?
5. What are the benefits of using the Rational Root Theorem?

**Tip**: When solving cubic equations, always test possible rational roots first to simplify the solution process.

Explore how to solve the cubic equation a^3 - 26a + 70 = 0 with step-by-step methods. Learn about the Rational Root Theorem, factoring, and numerical methods for solving cubic equations. Suitable for students in Grades 10-12.

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