Question

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

Answer 1

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

Explanation:

We can use polynomial long division or synthetic division to find the other factor(s) of the polynomial. Here, we'll use polynomial long division:

```

x^2 - x - 6

_____________

x - 1 | x^3 - 2x^2 - 5x + 6

- (x^3 - x^2)

------------

-x^2 - 5x

+ (x^2 - x)

----------

-6x + 6

- (-6x + 6)

----------

0

```

So, we have factored the polynomial as:

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

Now we need to factor the quadratic factor x^2 - x - 6. This can be factored as:

x^2 - x - 6 = (x - 3)(x + 2)

Therefore, we have fully factored the polynomial as:

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