Question

select the correct answer.

which function has zeros at -2, -1, and 4?

\\(f(x) = x(x - 2)(x - 1)(x - 4)\\)
\\(f(x) = (x - 2)(x - 1)(x + 4)^2\\)
\\(f(x) = (x + 2)^2(x + 1)(x - 4)\\)
\\(f(x) = (x + 2)(x + 1)(x + 4)\\)

Explanation:

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<reasoning>

Relate zeros to polynomial factors

The zeros of a polynomial function \(f(x)\) are the values of \(x\) for which \(f(x) = 0\). According to the Factor Theorem, if \(x = c\) is a zero of a polynomial function, then \((x - c)\) is a factor of the polynomial.

Write factors for the given zeros

We are given the zeros:

• \(x = -2\)
• \(x = -1\)
• \(x = 4\)

Using the Factor Theorem, we find the corresponding linear factors:

• For \(x = -2\), the factor is \((x - (-2)) = (x + 2)\).
• For \(x = -1\), the factor is \((x - (-1)) = (x + 1)\).
• For \(x = 4\), the factor is \((x - 4)\).

Form the general polynomial function

Any polynomial function with these zeros must be composed of these factors, potentially raised to positive integer powers (multiplicities). The general form is:
\[f(x) = a(x + 2)^p(x + 1)^q(x - 4)^r\]
where \(a
eq 0\) is a constant, and \(p, q, r \ge 1\).

Match with the given options

Let's evaluate the given choices:

• \(f(x) = x(x - 2)(x - 1)(x - 4)\): Zeros are \(0, 2, 1, 4\). (Incorrect)
• \(f(x) = (x - 2)(x - 1)(x + 4)^2\): Zeros are \(2, 1, -4\). (Incorrect)
• \(f(x) = (x + 2)^2(x + 1)(x - 4)\): Zeros are \(-2\) (multiplicity 2), \(-1\), and \(4\). (Correct)
• \(f(x) = (x + 2)(x + 1)(x + 4)\): Zeros are \(-2, -1, -4\). (Incorrect)

</reasoning>

<answer>
<mcq-option>(A) \(f(x) = x(x - 2)(x - 1)(x - 4)\)</mcq-option>
<mcq-option>(B) \(f(x) = (x - 2)(x - 1)(x + 4)^2\)</mcq-option>
<mcq-correct>(C) \(f(x) = (x + 2)^2(x + 1)(x - 4)\)</mcq-correct>
<mcq-option>(D) \(f(x) = (x + 2)(x + 1)(x + 4)\)</mcq-option>
</answer>

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</post_analysis>

Answer:

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<reasoning>

Relate zeros to polynomial factors

The zeros of a polynomial function \(f(x)\) are the values of \(x\) for which \(f(x) = 0\). According to the Factor Theorem, if \(x = c\) is a zero of a polynomial function, then \((x - c)\) is a factor of the polynomial.

Write factors for the given zeros

We are given the zeros:

• \(x = -2\)
• \(x = -1\)
• \(x = 4\)

Using the Factor Theorem, we find the corresponding linear factors:

• For \(x = -2\), the factor is \((x - (-2)) = (x + 2)\).
• For \(x = -1\), the factor is \((x - (-1)) = (x + 1)\).
• For \(x = 4\), the factor is \((x - 4)\).

Form the general polynomial function

Any polynomial function with these zeros must be composed of these factors, potentially raised to positive integer powers (multiplicities). The general form is:
\[f(x) = a(x + 2)^p(x + 1)^q(x - 4)^r\]
where \(a
eq 0\) is a constant, and \(p, q, r \ge 1\).

Match with the given options

Let's evaluate the given choices:

• \(f(x) = x(x - 2)(x - 1)(x - 4)\): Zeros are \(0, 2, 1, 4\). (Incorrect)
• \(f(x) = (x - 2)(x - 1)(x + 4)^2\): Zeros are \(2, 1, -4\). (Incorrect)
• \(f(x) = (x + 2)^2(x + 1)(x - 4)\): Zeros are \(-2\) (multiplicity 2), \(-1\), and \(4\). (Correct)
• \(f(x) = (x + 2)(x + 1)(x + 4)\): Zeros are \(-2, -1, -4\). (Incorrect)

</reasoning>

<answer>
<mcq-option>(A) \(f(x) = x(x - 2)(x - 1)(x - 4)\)</mcq-option>
<mcq-option>(B) \(f(x) = (x - 2)(x - 1)(x + 4)^2\)</mcq-option>
<mcq-correct>(C) \(f(x) = (x + 2)^2(x + 1)(x - 4)\)</mcq-correct>
<mcq-option>(D) \(f(x) = (x + 2)(x + 1)(x + 4)\)</mcq-option>
</answer>

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