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Answer :
To find all rational zeros of the polynomial function [tex]\( f(x) = 3x^5 - 23x^4 + 55x^3 - 45x^2 + 2x + 8 \)[/tex], we can use the Rational Root Theorem, which states that any rational root, [tex]\( \frac{p}{q} \)[/tex], of a polynomial is a factor of the constant term divided by a factor of the leading coefficient. In this case, our constant term is 8, and the leading coefficient is 3.
Step-by-Step Solution:
1. List all factors of the constant term (8):
Factors of 8: [tex]\( \pm 1, \pm 2, \pm 4, \pm 8 \)[/tex]
2. List all factors of the leading coefficient (3):
Factors of 3: [tex]\( \pm 1, \pm 3 \)[/tex]
3. Create all possible rational roots [tex]\( \frac{p}{q} \)[/tex]:
[tex]\[
\frac{\pm 1}{\pm 1}, \frac{\pm 1}{\pm 3}, \frac{\pm 2}{\pm 1}, \frac{\pm 2}{\pm 3}, \frac{\pm 4}{\pm 1}, \frac{\pm 4}{\pm 3}, \frac{\pm 8}{\pm 1}, \frac{\pm 8}{\pm 3}
\][/tex]
Simplifying, the possible rational roots are:
[tex]\[
\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3}, \pm 8, \pm \frac{8}{3}
\][/tex]
4. Test these possible roots in the polynomial [tex]\( f(x) \)[/tex].
We found that the polynomial has zeros at [tex]\( 1, 2, \)[/tex] and [tex]\( 4 \)[/tex]:
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3(1)^5 - 23(1)^4 + 55(1)^3 - 45(1)^2 + 2(1) + 8 = 3 - 23 + 55 - 45 + 2 + 8 = 0
\][/tex]
Hence, [tex]\( x = 1 \)[/tex] is a root.
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 3(2)^5 - 23(2)^4 + 55(2)^3 - 45(2)^2 + 2(2) + 8 = 96 - 368 + 440 - 180 + 4 + 8 = 0
\][/tex]
Hence, [tex]\( x = 2 \)[/tex] is a root.
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 3(4)^5 - 23(4)^4 + 55(4)^3 - 45(4)^2 + 2(4) + 8 = 3072 - 36864 + 14080 - 720 + 8 + 8 = 0
\][/tex]
Hence, [tex]\( x = 4 \)[/tex] is also a root.
5. List all the rational zeros we found:
[tex]\[
\boxed{1, 2, 4}
\][/tex]
These are all the rational zeros for the given polynomial function [tex]\( f(x) \)[/tex]. Thus, the set of all zeros of the given function is [tex]\( \{1, 2, 4\} \)[/tex].
Step-by-Step Solution:
1. List all factors of the constant term (8):
Factors of 8: [tex]\( \pm 1, \pm 2, \pm 4, \pm 8 \)[/tex]
2. List all factors of the leading coefficient (3):
Factors of 3: [tex]\( \pm 1, \pm 3 \)[/tex]
3. Create all possible rational roots [tex]\( \frac{p}{q} \)[/tex]:
[tex]\[
\frac{\pm 1}{\pm 1}, \frac{\pm 1}{\pm 3}, \frac{\pm 2}{\pm 1}, \frac{\pm 2}{\pm 3}, \frac{\pm 4}{\pm 1}, \frac{\pm 4}{\pm 3}, \frac{\pm 8}{\pm 1}, \frac{\pm 8}{\pm 3}
\][/tex]
Simplifying, the possible rational roots are:
[tex]\[
\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}, \pm 4, \pm \frac{4}{3}, \pm 8, \pm \frac{8}{3}
\][/tex]
4. Test these possible roots in the polynomial [tex]\( f(x) \)[/tex].
We found that the polynomial has zeros at [tex]\( 1, 2, \)[/tex] and [tex]\( 4 \)[/tex]:
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3(1)^5 - 23(1)^4 + 55(1)^3 - 45(1)^2 + 2(1) + 8 = 3 - 23 + 55 - 45 + 2 + 8 = 0
\][/tex]
Hence, [tex]\( x = 1 \)[/tex] is a root.
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 3(2)^5 - 23(2)^4 + 55(2)^3 - 45(2)^2 + 2(2) + 8 = 96 - 368 + 440 - 180 + 4 + 8 = 0
\][/tex]
Hence, [tex]\( x = 2 \)[/tex] is a root.
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 3(4)^5 - 23(4)^4 + 55(4)^3 - 45(4)^2 + 2(4) + 8 = 3072 - 36864 + 14080 - 720 + 8 + 8 = 0
\][/tex]
Hence, [tex]\( x = 4 \)[/tex] is also a root.
5. List all the rational zeros we found:
[tex]\[
\boxed{1, 2, 4}
\][/tex]
These are all the rational zeros for the given polynomial function [tex]\( f(x) \)[/tex]. Thus, the set of all zeros of the given function is [tex]\( \{1, 2, 4\} \)[/tex].
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