Posible factors of `x^(4)+x^(3)-7x^(2)-x+6` are

Options

A. `x+1`
B. `x+3`
C. `x-2`
D. All of these

Answer

To find the possible factors of the polynomial \( P(x) = x^4 + x^3 - 7x^2 - x + 6 \), we will use the trial and error method (also known as the Rational Root Theorem). This involves substituting possible rational roots into the polynomial to see if they yield a result of zero.

### Step 1: Identify possible rational roots
The possible rational roots can be the factors of the constant term (6) divided by the factors of the leading coefficient (1). The factors of 6 are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

### Step 2: Test \( x = 1 \)
Substituting \( x = 1 \):
\[ P(1) = 1^4 + 1^3 - 7 \cdot 1^2 - 1 + 6 = 1 + 1 - 7 - 1 + 6 = 0 \]
Since \( P(1) = 0 \), \( x - 1 \) is a factor.

### Step 3: Test \( x = 2 \)
Substituting \( x = 2 \):
\[ P(2) = 2^4 + 2^3 - 7 \cdot 2^2 - 2 + 6 = 16 + 8 - 28 - 2 + 6 = 0 \]
Since \( P(2) = 0 \), \( x - 2 \) is also a factor.

### Step 4: Factor the polynomial
Now that we have two factors, \( x - 1 \) and \( x - 2 \), we can divide the polynomial \( P(x) \) by \( (x - 1)(x - 2) \).

First, we multiply the factors:
\[ (x - 1)(x - 2) = x^2 - 3x + 2 \]

### Step 5: Perform polynomial long division
Now we will divide \( P(x) \) by \( x^2 - 3x + 2 \).

1. Divide \( x^4 \) by \( x^2 \) to get \( x^2 \).
2. Multiply \( x^2 \) by \( (x^2 - 3x + 2) \) to get \( x^4 - 3x^3 + 2x^2 \).
3. Subtract this from \( P(x) \):
\[ (x^4 + x^3 - 7x^2 - x + 6) - (x^4 - 3x^3 + 2x^2) = 4x^3 - 9x^2 - x + 6 \]
4. Now divide \( 4x^3 \) by \( x^2 \) to get \( 4x \).
5. Multiply \( 4x \) by \( (x^2 - 3x + 2) \) to get \( 4x^3 - 12x^2 + 8x \).
6. Subtract this from the previous result:
\[ (4x^3 - 9x^2 - x + 6) - (4x^3 - 12x^2 + 8x) = 3x^2 - 9x + 6 \]
7. Now divide \( 3x^2 \) by \( x^2 \) to get \( 3 \).
8. Multiply \( 3 \) by \( (x^2 - 3x + 2) \) to get \( 3x^2 - 9x + 6 \).
9. Subtract this:
\[ (3x^2 - 9x + 6) - (3x^2 - 9x + 6) = 0 \]

### Step 6: Factor the quotient
The quotient we obtained is \( x^2 - 3x + 2 \). We can factor this further:
\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]

### Step 7: Combine all factors
Thus, the complete factorization of the polynomial \( P(x) \) is:
\[ P(x) = (x - 1)^2 (x - 2)^2 \]

### Conclusion
The possible factors of \( x^4 + x^3 - 7x^2 - x + 6 \) are \( (x - 1)^2 \) and \( (x - 2)^2 \).