Decompose the polynomial x^3−2x^2−5x+6 into its linear factors, given that x−1 is a factor.

Answer 1

Since x - 1 is a factor of the polynomial, we know that (x - 1) divides evenly into the polynomial when we perform polynomial long division.

Performing polynomial long division:
___________________
(x - 1) | x^3 - 2x^2 - 5x + 6
- (x^3 - x^2)
------------
- x^2 - 5x
+ (x^2 - x)
------------
- 4x + 6
+ (4x - 4)
------------
2

After performing polynomial long division, we have:
x^3 - 2x^2 - 5x + 6 = (x - 1)(x^2 - x - 4) + 2

So, the polynomial x^3 - 2x^2 - 5x + 6 can be factored as:
(x - 1)(x^2 - x - 4) + 2