Find the value of a for which (x+a) is factor of x³+ax²+2x+a-1

We need to determine the value of \( a \) such that \( (x + a) \) is a factor of the polynomial \( x^3 + ax^2 + 2x + a - 1 \).

### Approach:
If \( (x + a) \) is a factor of \( x^3 + ax^2 + 2x + a - 1 \), then substituting \( x = -a \) into the polynomial should yield 0 (since \( x + a = 0 \)).

Let the polynomial be \( f(x) = x^3 + ax^2 + 2x + a - 1 \).

Now, substitute \( x = -a \) into \( f(x) \):

\[
f(-a) = (-a)^3 + a(-a)^2 + 2(-a) + a - 1
\]

Simplifying:

\[
f(-a) = -a^3 + a \cdot a^2 - 2a + a - 1
\]
\[
f(-a) = -a^3 + a^3 - a + a - 1
\]
\[
f(-a) = -1
\]

This result shows that \( f(-a) = -1 \), which means that \( x + a \) is **not** a factor for any value of \( a \) because \( f(-a) \neq 0 \).

Thus, there is **no value of \( a \)** that makes \( (x + a) \) a factor of \( x^3 + ax^2 + 2x + a - 1 \).

Would you like further explanation, or do you have any additional questions?

Here are some related questions you might explore:
1. How to factor cubic polynomials?
2. What is the remainder theorem?
3. How does synthetic division work for polynomials?
4. What is the role of roots in polynomial factorization?
5. Can any other linear factor exist for this polynomial?

**Tip:** Always verify your solution by plugging values back into the original equation.

Learn how to determine the value of 'a' for which (x+a) is a factor of the cubic polynomial x³+ax²+2x+a-1. This problem explores key algebraic concepts including polynomial factorization and the Remainder Theorem. Suitable for students in Grades 10-12.

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