Question
Find all the zeros of the function
y = x^3 + 6x^2 + x + 6
y = x^3 + 6x^2 + x + 6
Answers
Answered by
GPT 3.5
To find the zeros of the function, we need to find the values of x for which y = 0.
We can use synthetic division to test potential zeros. Let's start with x = -1, which is a common choice because it is a factor of the constant term (6).
-1 | 1 6 1 6
| -1 -5 4
-------------
1 5 -4 10
The remainder is 10, so x = -1 is not a zero. Let's try x = 1.
1 | 1 6 1 6
| 1 7 8
------------
1 7 8 14
The remainder is 14, so x = 1 is not a zero either. Now let's try x = -2.
-2 | 1 6 1 6
| -2 -8 14
-------------
1 4 -7 20
The remainder is 20, so x = -2 is not a zero.
Since we have tried all the potential rational zeros, none of them are zeros of the function. Therefore, the function has no real zeros.
We can use synthetic division to test potential zeros. Let's start with x = -1, which is a common choice because it is a factor of the constant term (6).
-1 | 1 6 1 6
| -1 -5 4
-------------
1 5 -4 10
The remainder is 10, so x = -1 is not a zero. Let's try x = 1.
1 | 1 6 1 6
| 1 7 8
------------
1 7 8 14
The remainder is 14, so x = 1 is not a zero either. Now let's try x = -2.
-2 | 1 6 1 6
| -2 -8 14
-------------
1 4 -7 20
The remainder is 20, so x = -2 is not a zero.
Since we have tried all the potential rational zeros, none of them are zeros of the function. Therefore, the function has no real zeros.