find the value of a for which (x-a) is factor of x-ax+2a+a-1

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

### Step 1: Simplify the given expression.
The expression is:
\[
x - ax + 2a + a - 1
\]
We can combine like terms:
\[
x(1 - a) + 3a - 1
\]

Thus, the simplified form of the expression is:
\[
x(1 - a) + 3a - 1
\]

### Step 2: Apply the factor condition.
For \( (x - a) \) to be a factor of this expression, the remainder when we substitute \( x = a \) into the expression must be zero. So, we substitute \( x = a \) into the expression:
\[
a(1 - a) + 3a - 1 = 0
\]

### Step 3: Solve for \( a \).
Now, solve the equation:
\[
a(1 - a) + 3a - 1 = 0
\]
Expanding the terms:
\[
a - a^2 + 3a - 1 = 0
\]
Combine like terms:
\[
-a^2 + 4a - 1 = 0
\]
This is a quadratic equation:
\[
a^2 - 4a + 1 = 0
\]

### Step 4: Use the quadratic formula.
To solve \( a^2 - 4a + 1 = 0 \), we use the quadratic formula:
\[
a = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(1)}}{2(1)}
\]
Simplifying:
\[
a = \frac{4 \pm \sqrt{16 - 4}}{2}
\]
\[
a = \frac{4 \pm \sqrt{12}}{2}
\]
\[
a = \frac{4 \pm 2\sqrt{3}}{2}
\]
\[
a = 2 \pm \sqrt{3}
\]

### Final Answer:
The value of \( a \) is \( a = 2 + \sqrt{3} \) or \( a = 2 - \sqrt{3} \).

Do you want further details or have any questions?

Here are 5 related questions to expand the topic:
1. How can we verify that \( x - a \) is indeed a factor of the expression for these values of \( a \)?
2. How does the factorization process change for more complex polynomial expressions?
3. What is the significance of the quadratic formula in finding roots of quadratic equations?
4. Can this method be applied to higher-degree polynomials?
5. How does synthetic division work for testing if \( (x - a) \) is a factor of a polynomial?

**Tip:** Always verify by substituting the solution back into the original equation or expression to check correctness.

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

This problem focuses on finding the value of a for which (x-a) is a factor of the polynomial expression x-ax+2a+a-1. Using key algebraic techniques like the quadratic formula and factor theorem, we derive the values of a. Suitable for Grades 9-11.

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