Use factor theorem to find one factor of polynomial y= x^3 + 2x^2 -5x -6 ?

To find a factor of the polynomial y = x^3 + 2x^2 - 5x - 6 using the factor theorem, we need to find a value of x that makes the polynomial equal to zero.

Let's set y = 0 and solve for x:

0 = x^3 + 2x^2 - 5x - 6

Now, we need to find a value of x that satisfies this equation. One way to do this is to try different values of x until we find one that makes the equation equal to zero.

By trying x = 1, we get:

0 = 1^3 + 2(1)^2 - 5(1) - 6
0 = 1 + 2 - 5 - 6
0 = 0

Since x = 1 makes the polynomial equal to zero, x - 1 is a factor of the polynomial y = x^3 + 2x^2 - 5x - 6.

Therefore, one factor of the polynomial is (x - 1).

Step 1: Determine a possible factor

To use the factor theorem, we need to find a number that, when plugged into the polynomial, makes the result 0. Let's try x = -1:

y = (-1)^3 + 2(-1)^2 - 5(-1) - 6
= -1 + 2 + 5 - 6
= 0

Therefore, x + 1 is a possible factor of the polynomial.

Step 2: Divide the polynomial by the factor

We can use synthetic division to divide the polynomial by x + 1:

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

The remainder is 0, which confirms that x + 1 is a factor of the polynomial.

Therefore, one factor of the polynomial y = x^3 + 2x^2 - 5x - 6 is x + 1.