Determine whether x-1 is a factor of `x^6-x^5+x^4+x^3-x^2-x+1` or not?

Answer

To determine whether \( x - 1 \) is a factor of the polynomial \( f(x) = x^6 - x^5 + x^4 + x^3 - x^2 - x + 1 \), we can use the Factor Theorem. According to the Factor Theorem, if \( x - c \) is a factor of \( f(x) \), then \( f(c) = 0 \).

### Step 1: Define the polynomial
Let \( f(x) = x^6 - x^5 + x^4 + x^3 - x^2 - x + 1 \).

### Step 2: Substitute \( x = 1 \) into the polynomial
We need to evaluate \( f(1) \):
\[ f(1) = 1^6 - 1^5 + 1^4 + 1^3 - 1^2 - 1 + 1 \]

### Step 3: Calculate each term
Calculating each term:
- \( 1^6 = 1 \)
- \( 1^5 = 1 \)
- \( 1^4 = 1 \)
- \( 1^3 = 1 \)
- \( 1^2 = 1 \)

Now substituting these values back into the equation:
\[ f(1) = 1 - 1 + 1 + 1 - 1 - 1 + 1 \]

### Step 4: Simplify the expression
Now, simplify the expression step by step:
\[ f(1) = (1 - 1) + (1 + 1) - (1 + 1) = 0 + 2 - 2 = 0 \]

### Step 5: Conclusion
Since \( f(1) = 0 \), according to the Factor Theorem, \( x - 1 \) is a factor of the polynomial \( f(x) \).

### Final Answer
Thus, \( x - 1 \) is a factor of \( x^6 - x^5 + x^4 + x^3 - x^2 - x + 1 \).

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