How can we factor the cubic polynomial a3−3a+2*a*3−3*a*+2?

To factor the cubic polynomial \( a^3 - 3a + 2 \), we can use the Rational Root Theorem, which suggests that any rational root, expressed as \( \frac{p}{q} \), must be a factor of the constant term divided by a factor of the leading coefficient. Here, the constant term is 2, and the leading coefficient is 1.

### Steps to Factor the Polynomial

1. **Identify possible rational roots:**
   The possible rational roots are the factors of 2 (constant term) divided by the factors of 1 (leading coefficient). So, the possible roots are \( \pm 1 \) and \( \pm 2 \).

2. **Test the possible roots:**
   Substitute these values into the polynomial to find actual roots.

   - For \( a = 1 \):
     \[ 1^3 - 3(1) + 2 = 1 - 3 + 2 = 0 \]
     So, \( a = 1 \) is a root.

   - For \( a = -1 \):
     \[ (-1)^3 - 3(-1) + 2 = -1 + 3 + 2 = 4 \neq 0 \]
     \( a = -1 \) is not a root.

   - For \( a = 2 \):
     \[ 2^3 - 3(2) + 2 = 8 - 6 + 2 = 4 \neq 0 \]
     \( a = 2 \) is not a root.

   - For \( a = -2 \):
     \[ (-2)^3 - 3(-2) + 2 = -8 + 6 + 2 = 0 \]
     So, \( a = -2 \) is a root.

3. **Factor the polynomial using the found roots:**
   Since \( a = 1 \) and \( a = -2 \) are roots, the polynomial can be factored as:
   \[ (a - 1)(a + 2) \]

   We can then factorize the remaining quadratic equation. Using synthetic division or polynomial division, we divide \( a^3 - 3a + 2 \) by \( (a - 1)(a + 2) \) to find the other factors.

4. **Find the complete factorization:**
   After division, we get the other factor, \( (a^2 + a - 2) \). Factoring this further:
   \[ a^2 + a - 2 = (a - 1)(a + 2) \]

So, the complete factorization of the polynomial is:
\[ (a - 1)(a + 2)(a - 1) = (a - 1)^2(a + 2) \]

### Summary
The cubic polynomial \( a^3 - 3a + 2 \) factors into:
\[ (a - 1)^2(a + 2) \]

Would you like any additional information or clarifications?

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Here are some related questions:
1. How do we confirm that all the roots of the polynomial \( a^3 - 3a + 2 \) are found?
2. What is the significance of the factorization in solving polynomial equations?
3. How can we use synthetic division to find roots of a polynomial?
4. How does the graph of \( a^3 - 3a + 2 \) change near its roots?
5. What are some common methods for finding roots of cubic equations?
6. Can the factor theorem be used to find roots of \( a^3 - 3a + 2 \)?
7. How does the discriminant of the quadratic factor \( a^2 + a - 2 \) relate to the nature of its roots?
8. What are the possible configurations of real and complex roots for cubic polynomials?

**Tip:** When factoring polynomials, always check the work by expanding the factors to ensure they equal the original polynomial.

Learn how to factor the cubic polynomial a^3 - 3a + 2 using the Rational Root Theorem and synthetic division. This problem involves finding rational roots and using them to factorize the polynomial. Suitable for advanced high school students and college algebra courses.

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