To find the polynomial that has zeros at \( -4 \) and \( -6 \), we can use the factored form of a polynomial related to its zeros. Specifically, if \( r_1 \) and \( r_2 \) are the zeros, the polynomial can be expressed as:
\[ y = a(x - r_1)(x - r_2) \]
In this case, the zeros are \( r_1 = -4 \) and \( r_2 = -6 \). Substituting these values into the equation gives:
\[ y = a(x + 4)(x + 6) \]
Next, we can expand this expression:
\[ y = a[(x + 4)(x + 6)] = a[x^2 + 6x + 4x + 24] = a[x^2 + 10x + 24] \]
To find the simplest polynomial, we set \( a = 1 \):
\[ y = x^2 + 10x + 24 \]
Now let's check the given options:
- A: \( y = x^2 - 10x - 24 \)
- B: \( y = x^2 + 10x + 24 \)
- C: \( y = x^2 + 10x - 24 \)
- D: \( y = x^2 - 10x + 24 \)
The correct function, defined by a polynomial with zeros at \( -4 \) and \( -6 \), is:
B: \( y = x^2 + 10x + 24 \).