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Answer :
To find the potential rational zeros of the polynomial function [tex]\( f(x) = 2x^{11} - x^9 + 7x^8 + 116 \)[/tex] using the Rational Zeros Theorem, you follow these steps:
1. Identify the Constant and Leading Coefficient:
- The constant term (the term without [tex]\( x \)[/tex]) is 116.
- The leading coefficient (the coefficient of the term with the highest power of [tex]\( x \)[/tex]) is 2.
2. List the Factors:
- Find all the factors of the constant term, 116. These are: [tex]\( \pm 1, \pm 2, \pm 4, \pm 29, \pm 58, \pm 116 \)[/tex].
- Find all the factors of the leading coefficient, 2. These are: [tex]\( \pm 1, \pm 2 \)[/tex].
3. Form Possible Rational Zeros:
- According to the Rational Zeros Theorem, the potential rational zeros are given by [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient.
- By combining these factors, we form the potential rational zeros: [tex]\[
\pm 1, \pm 2, \pm 4, \pm 29, \pm 58, \pm 116, \pm \frac{1}{2}, \pm \frac{2}{2} = \pm 1, \pm \frac{4}{2} = \pm 2, \pm \frac{29}{2}, \pm \frac{58}{2}, \pm \frac{116}{2}
\][/tex]
4. Simplify and List the Zeros:
- Notice that some fractions simplify to integers already listed, so we narrow down our list to these distinct values: [tex]\[
-1, 1, -2, 2, -4, 4, -29, 29, -58, 58, -116, 116, -\frac{1}{2}, \frac{1}{2}
\][/tex]
After following these steps, the correct answer choice is:
D. [tex]\(-1, 1, -2, 2, -4, 4, -29, 29, -58, 58, -116, 116, -\frac{1}{2}, \frac{1}{2}\)[/tex]
1. Identify the Constant and Leading Coefficient:
- The constant term (the term without [tex]\( x \)[/tex]) is 116.
- The leading coefficient (the coefficient of the term with the highest power of [tex]\( x \)[/tex]) is 2.
2. List the Factors:
- Find all the factors of the constant term, 116. These are: [tex]\( \pm 1, \pm 2, \pm 4, \pm 29, \pm 58, \pm 116 \)[/tex].
- Find all the factors of the leading coefficient, 2. These are: [tex]\( \pm 1, \pm 2 \)[/tex].
3. Form Possible Rational Zeros:
- According to the Rational Zeros Theorem, the potential rational zeros are given by [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient.
- By combining these factors, we form the potential rational zeros: [tex]\[
\pm 1, \pm 2, \pm 4, \pm 29, \pm 58, \pm 116, \pm \frac{1}{2}, \pm \frac{2}{2} = \pm 1, \pm \frac{4}{2} = \pm 2, \pm \frac{29}{2}, \pm \frac{58}{2}, \pm \frac{116}{2}
\][/tex]
4. Simplify and List the Zeros:
- Notice that some fractions simplify to integers already listed, so we narrow down our list to these distinct values: [tex]\[
-1, 1, -2, 2, -4, 4, -29, 29, -58, 58, -116, 116, -\frac{1}{2}, \frac{1}{2}
\][/tex]
After following these steps, the correct answer choice is:
D. [tex]\(-1, 1, -2, 2, -4, 4, -29, 29, -58, 58, -116, 116, -\frac{1}{2}, \frac{1}{2}\)[/tex]
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