Use the long division method and factorization method to find the two factor of polynomial y= x^3 + 2x^2 - 5x -6 ?

To find the factors of the polynomial y = x^3 + 2x^2 - 5x - 6, we can use both the long division method and the factorization method.

1. Long Division Method:
First, we can use the long division method to divide the polynomial by a linear factor (x - a) to find the other factor.

Let's assume that (x - 1) is a factor of the polynomial. We can perform long division as follows:

x^2 + 3x + 6
_______________________
x - 1 | x^3 + 2x^2 - 5x - 6
- (x^3 - x^2)
_______________
3x^2 - 5x
- (3x^2 - 3x)
___________
-2x - 6
- (-2x + 2)
__________
-8

Since the remainder is not zero, (x - 1) is not a factor of the polynomial. We can try another linear factor.

2. Factorization Method:
To find the factors of the polynomial, we can factorize it by grouping the terms:

y = x^3 + 2x^2 - 5x - 6
= x^2(x + 2) - 5(x + 2)
= (x^2 - 5)(x + 2)
= (x - √5)(x + √5)(x + 2)

Therefore, the two factors of the polynomial y = x^3 + 2x^2 - 5x - 6 are (x - √5), (x + √5), and (x + 2).

Long Division Method:

Dividing y = x^3 + 2x^2 - 5x - 6 by x - 2:

```
x^2 + 4x + 2
x - 2 | x^3 + 2x^2 - 5x - 6
- (x^3 - 2x^2)
--------
4x^2 - 5x
- (4x^2 - 8x)
--------
5x - 6
- (5x - 10)
--------
16
```

Therefore, y = x^3 + 2x^2 - 5x - 6 = (x - 2)(x^2 + 4x + 2) + 16.

Factorization Method:

1. Find two numbers that add up to -5 (the coefficient of the middle term) and multiply to -6 (the constant term). These numbers are -6 and 1.

2. Rewrite the middle term -5x as -6x + x.

3. Factor by grouping:

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

Since x^2 + x - 3 is a quadratic trinomial that cannot be factored over the rational numbers, the two factors of y are:

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