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Which function represents a function with zeros at [tex]$-3, -1, 0$[/tex], and [tex]$6$[/tex]?

A. [tex]$y=(x-6)(x+1)(x+3)$[/tex]
B. [tex]$y=x(x-3)(x-1)(x+6)$[/tex]
C. [tex]$y=x(x-6)(x+1)(x+3)$[/tex]
D. [tex]$y=(x-3)(x-1)(x+6)$[/tex]

Sagot :

GinnyAnswer
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]
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