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Answer :
Sure, let's use the Rational Zeros Theorem to find the potential rational zeros of the polynomial [tex]\( f(x) = 2x^{11} - x^9 + 2x^8 + 116 \)[/tex].
### Step 1: Identify the Constant Term and Leading Coefficient
- Constant term: The constant term in the polynomial is [tex]\( 116 \)[/tex].
- Leading coefficient: The leading coefficient (the coefficient of the highest degree term) is [tex]\( 2 \)[/tex].
### Step 2: Determine Factors
- Factors of 116: The integer factors (both positive and negative) of 116 are:
[tex]\(-1, 1, -2, 2, -4, 4, -29, 29, -58, 58, -116, 116 \)[/tex].
- Factors of 2: The integer factors (both positive and negative) of 2 are:
[tex]\(-1, 1, -2, 2 \)[/tex].
### Step 3: Form Potential Rational Zeros
Using the factors from above, we form the potential rational zeros by taking each factor of the constant term and dividing it by each factor of the leading coefficient. The potential zeros are:
[tex]\[
\pm \frac{1}{1}, \pm \frac{2}{1}, \pm \frac{4}{1}, \pm \frac{29}{1}, \pm \frac{58}{1}, \pm \frac{116}{1},
\pm \frac{1}{2}, \pm \frac{2}{2}, \pm \frac{4}{2}, \pm \frac{29}{2}, \pm \frac{58}{2}, \pm \frac{116}{2}
\][/tex]
Simplifying these, we get:
- [tex]\(-1, 1, -2, 2, -4, 4, -29, 29, -58, 58, -116, 116\)[/tex]
- [tex]\(-\frac{1}{2}, \frac{1}{2}, -\frac{29}{2}, \frac{29}{2}, -\frac{58}{2}, \frac{58}{2}, -\frac{116}{2}, \frac{116}{2}\)[/tex]
Simplifying the fractions:
- [tex]\(-\frac{58}{2} = -29\)[/tex] and [tex]\(\frac{58}{2} = 29\)[/tex]
- [tex]\(-\frac{116}{2} = -58\)[/tex] and [tex]\(\frac{116}{2} = 58\)[/tex]
### Step 4: List the Unique Potential Rational Zeros
After simplifying and removing duplicates, the set of potential rational zeros includes:
- [tex]\(-1, 1, -2, 2, -4, 4, -29, 29, -58, 58, -116, 116\)[/tex]
- [tex]\(-\frac{1}{2}, \frac{1}{2}, -\frac{29}{2}, \frac{29}{2}\)[/tex]
Now, let's match this list with the given options:
- Option A: Includes [tex]\(-\frac{1}{116}, \frac{1}{116}\)[/tex] which are not needed here.
- Option B: It correctly matches but misses some potential zeros and introduces incorrect ones.
- Option C: Exactly matches without unnecessary additions.
- Option D: Has numbers like [tex]\(-58, 58, -116, 116\)[/tex] but also includes incorrect fractions like [tex]\(-\frac{29}{2}, \frac{29}{2}\)[/tex].
Upon comparison, Option C correctly matches the potential rational zeros we listed, without any incorrect entries. Therefore, Option C is the correct list of potential rational zeros.
### Step 1: Identify the Constant Term and Leading Coefficient
- Constant term: The constant term in the polynomial is [tex]\( 116 \)[/tex].
- Leading coefficient: The leading coefficient (the coefficient of the highest degree term) is [tex]\( 2 \)[/tex].
### Step 2: Determine Factors
- Factors of 116: The integer factors (both positive and negative) of 116 are:
[tex]\(-1, 1, -2, 2, -4, 4, -29, 29, -58, 58, -116, 116 \)[/tex].
- Factors of 2: The integer factors (both positive and negative) of 2 are:
[tex]\(-1, 1, -2, 2 \)[/tex].
### Step 3: Form Potential Rational Zeros
Using the factors from above, we form the potential rational zeros by taking each factor of the constant term and dividing it by each factor of the leading coefficient. The potential zeros are:
[tex]\[
\pm \frac{1}{1}, \pm \frac{2}{1}, \pm \frac{4}{1}, \pm \frac{29}{1}, \pm \frac{58}{1}, \pm \frac{116}{1},
\pm \frac{1}{2}, \pm \frac{2}{2}, \pm \frac{4}{2}, \pm \frac{29}{2}, \pm \frac{58}{2}, \pm \frac{116}{2}
\][/tex]
Simplifying these, we get:
- [tex]\(-1, 1, -2, 2, -4, 4, -29, 29, -58, 58, -116, 116\)[/tex]
- [tex]\(-\frac{1}{2}, \frac{1}{2}, -\frac{29}{2}, \frac{29}{2}, -\frac{58}{2}, \frac{58}{2}, -\frac{116}{2}, \frac{116}{2}\)[/tex]
Simplifying the fractions:
- [tex]\(-\frac{58}{2} = -29\)[/tex] and [tex]\(\frac{58}{2} = 29\)[/tex]
- [tex]\(-\frac{116}{2} = -58\)[/tex] and [tex]\(\frac{116}{2} = 58\)[/tex]
### Step 4: List the Unique Potential Rational Zeros
After simplifying and removing duplicates, the set of potential rational zeros includes:
- [tex]\(-1, 1, -2, 2, -4, 4, -29, 29, -58, 58, -116, 116\)[/tex]
- [tex]\(-\frac{1}{2}, \frac{1}{2}, -\frac{29}{2}, \frac{29}{2}\)[/tex]
Now, let's match this list with the given options:
- Option A: Includes [tex]\(-\frac{1}{116}, \frac{1}{116}\)[/tex] which are not needed here.
- Option B: It correctly matches but misses some potential zeros and introduces incorrect ones.
- Option C: Exactly matches without unnecessary additions.
- Option D: Has numbers like [tex]\(-58, 58, -116, 116\)[/tex] but also includes incorrect fractions like [tex]\(-\frac{29}{2}, \frac{29}{2}\)[/tex].
Upon comparison, Option C correctly matches the potential rational zeros we listed, without any incorrect entries. Therefore, Option C is the correct list of potential rational zeros.
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