To determine which function represents a polynomial with zeros at [tex]\(-3\)[/tex], [tex]\(-1\)[/tex], [tex]\(0\)[/tex], and [tex]\(6\)[/tex], we need to check which polynomial equations have these specific roots.
1. Analyzing Option A: [tex]\(y = (x-6)(x+1)(x+3)\)[/tex]
This function can be expanded to find its roots:
[tex]\[ y = (x-6)(x+1)(x+3) \][/tex]
- The roots are the values of [tex]\(x\)[/tex] that make each factor equal to zero.
- The roots here are found by solving:
[tex]\[
x - 6 = 0 \implies x = 6 \\
x + 1 = 0 \implies x = -1 \\
x + 3 = 0 \implies x = -3
\][/tex]
- This function has zeros at [tex]\(6\)[/tex], [tex]\(-1\)[/tex], and [tex]\(-3\)[/tex], but it's missing [tex]\(0\)[/tex].
2. Analyzing Option B: [tex]\(y = x(x-3)(x-1)(x+6)\)[/tex]
This function can be expanded to find its roots:
[tex]\[ y = x(x-3)(x-1)(x+6) \][/tex]
- The roots are found by solving:
[tex]\[
x = 0 \\
x-3 = 0 \implies x = 3 \\
x-1 = 0 \implies x = 1 \\
x + 6 = 0 \implies x = -6
\][/tex]
- This function has zeros at [tex]\(0\)[/tex], [tex]\(3\)[/tex], [tex]\(1\)[/tex], and [tex]\(-6\)[/tex]. These do not match [tex]\(-3\)[/tex], [tex]\(-1\)[/tex], [tex]\(0\)[/tex], and [tex]\(6\)[/tex].
3. Analyzing Option C: [tex]\(y = x(x-6)(x+1)(x+3)\)[/tex]
This function can be expanded to find its roots:
[tex]\[ y = x(x-6)(x+1)(x+3) \][/tex]
- The roots are found by solving:
[tex]\[
x = 0 \\
x - 6 = 0 \implies x = 6 \\
x + 1 = 0 \implies x = -1 \\
x + 3 = 0 \implies x = -3
\][/tex]
- This function has zeros at [tex]\(0\)[/tex], [tex]\(6\)[/tex], [tex]\(-1\)[/tex], and [tex]\(-3\)[/tex], which correctly match the given zeros.
4. Analyzing Option D: [tex]\(y = (x-3)(x-1)(x+6)\)[/tex]
This function can be expanded to find its roots:
[tex]\[ y = (x-3)(x-1)(x+6) \][/tex]
- The roots are found by solving:
[tex]\[
x-3 = 0 \implies x = 3 \\
x-1 = 0 \implies x = 1 \\
x + 6 = 0 \implies x = -6
\][/tex]
- This function has zeros at [tex]\(3\)[/tex], [tex]\(1\)[/tex], and [tex]\(-6\)[/tex], but it does not include [tex]\(0\)[/tex], [tex]\(-3\)[/tex], and [tex]\(-1\)[/tex].
Upon reviewing each option, we conclude that the correct polynomial function with zeros at [tex]\(-3\)[/tex], [tex]\(-1\)[/tex], [tex]\(0\)[/tex], and [tex]\(6\)[/tex] is:
Option C: [tex]\(y = x(x-6)(x+1)(x+3)\)[/tex]
