Question

Decompose the polynomial x3−2x2−5x+6 into its linear factors, given that x−1 is a factor. (1 point)

Answer 1

To decompose the polynomial \(x^3 - 2x^2 - 5x + 6\) into its linear factors, given that \(x - 1\) is a factor, we can use polynomial long division or synthetic division.

### Step 1: Synthetic Division

Since \(x - 1\) is a factor, we can use synthetic division with \(1\):

1. Write the coefficients of the polynomial: \(1, -2, -5, 6\).

2. Set up the synthetic division:

```
1 | 1 -2 -5 6
| 1 -1 -6
-----------------
1 -1 -6 0
```

The bottom row shows the coefficients of the quotient polynomial, which is \(1x^2 - 1x - 6\).

### Step 2: Factor the Quotient

Now, we need to factor the quadratic \(x^2 - x - 6\):

To factor \(x^2 - x - 6\), we look for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\).

Thus:

\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

### Step 3: Combine the Factors

The complete factorization of the original polynomial is:

\[
x^3 - 2x^2 - 5x + 6 = (x - 1)(x - 3)(x + 2)
\]

### Final Answer

The linear factors of the polynomial \(x^3 - 2x^2 - 5x + 6\) are:

\[
(x - 1)(x - 3)(x + 2)
\]