Respuesta :
The correct option is (2) [tex]\frac{8}{5}[/tex] and (3) [tex]\frac{1}{6}[/tex].
The factor of equation f(x) = 60[tex]x^{4}[/tex] + 86[tex]x^{3}[/tex] – 46[tex]x^{2}[/tex] – 43[tex]x[/tex] + 8 are [tex]\frac{8}{5}[/tex] and [tex]\frac{1}{6}[/tex].
Rational Root Theorem:
The rational root theorem, also referred to as the rational zero theorem, is a potent mathematical technique used to identify all potential rational roots of polynomial equations of order 3 and above.
For the given polynomial, the roots are given as;
[tex]$P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0}$\\ \\$\pm \frac{\text { factors of } a_{0}}{\text { factors of } a_{n}}$.[/tex]
Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 and factors of 8 are 1, 2, 4 and 8.
Out of the given options, only [tex]\frac{8}{5}[/tex] and [tex]\frac{1}{6}[/tex] can be written in the form:
[tex]\begin{aligned}&\frac{8}{5}=\frac{\text { a factor of } 8}{\text { a factor of } 60} \\&\frac{1}{6}=\frac{\text { a factor of } 8}{\text { a factor of } 60}\end{aligned}[/tex]
Therefore, the rational roots of polynomial f(x) = 60[tex]x^{4}[/tex] + 86[tex]x^{3}[/tex] – 46[tex]x^{2}[/tex] – 43[tex]x[/tex] + 8 are [tex]\frac{8}{5}[/tex] and [tex]\frac{1}{6}[/tex].
To know more about Rational Root Theorem, here
https://brainly.com/question/10937559
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The complete question is-
Which is a factor of f(x) = 60[tex]x^{4}[/tex] + 86[tex]x^{3}[/tex] – 46[tex]x^{2}[/tex] – 43[tex]x[/tex] + 8? use the rational root theorem to help you find your answer.
- x – 6 => 6
- 5x – 8 => [tex]\frac{8}{5}[/tex]
- 6x – 1 => [tex]\frac{1}{6}[/tex]
- 8x + 5 => [tex]\frac{-5}{8}[/tex]
