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

We are given the expression \( x^3 + ax^2 + 2x + a - 1 \), and we need to find the value of \( a \) such that \( (x + a) \) is a factor of the given polynomial.

If \( (x + a) \) is a factor, then by the factor theorem, substituting \( x = -a \) into the polynomial should result in zero.

Let's substitute \( x = -a \) into the polynomial \( x^3 + ax^2 + 2x + a - 1 \):

\[
(-a)^3 + a(-a)^2 + 2(-a) + a - 1 = 0
\]

Simplifying this:

\[
-a^3 + a(a^2) - 2a + a - 1 = 0
\]

\[
-a^3 + a^3 - 2a + a - 1 = 0
\]

Now, the \( -a^3 + a^3 \) terms cancel out, and we are left with:

\[
-2a + a - 1 = 0
\]

\[
-a - 1 = 0
\]

Solving for \( a \):

\[
-a = 1 \quad \Rightarrow \quad a = -1
\]

Thus, the value of \( a \) is \( \boxed{-1} \).

Would you like more details or have any questions? Here are 5 related questions for further practice:

1. What is the factor theorem and how does it apply to polynomials?
2. How do you determine the factors of a cubic polynomial?
3. What other values of \( a \) could be considered if the form of the factor changed?
4. How do you fully factorize a cubic polynomial like \( x^3 - x^2 + 2x - 2 \)?
5. What is synthetic division and how can it help in factoring polynomials?

**Tip:** Always check your solution by substituting the found value back into the original polynomial.

Explore how to find the value of a for which (x + a) is a factor of the cubic polynomial x³ + ax² + 2x + a - 1. This problem involves applying the factor theorem and solving polynomial equations, suitable for students in Grades 9-11.

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