ANSWER
The possible rational roots are:
[tex]\pm\frac{1}{2},\pm2,\pm \frac{5}{2}[/tex]
EXPLANATION
According to the rational root theorem, the possible rational roots of the polynomial,
[tex]4x^3+9x^2-x+10=0[/tex]
are all the possible factors of the constant term divided by all the possible factors of the coefficient of the highest degree of the polynomial.
Therefore we examine the numerator and denominator of the options provided to see if they are factors of 10 and 4 respectively.
For
[tex]\pm\frac{1}{2}[/tex], we can see that 1 is a factor of 10 and 2 is a factor of 4, hence it is a possible rational root.
The same thing applies to [tex]\pm2,\pm \frac{5}{2}[/tex] also.
As for
[tex]\pm \frac{2}{5}[/tex], 2 is a factor of 10 but 5 is not a factor of 2, hence it is not a possible rational root.
The values of the possible rational roots of 4x³ + 9x² - x + 10 = 0 among the options are;
±½, ±2, ±5/2
The rational root theorem states that if a polynomial has any rational roots, then the rational roots must be of the form;
±(factors of the coefficient of the constant term/factors of coefficient of the term with the highest degree)
4x³ + 9x² - x + 10 = 0
Thus, the rational roots could be;
±(½, 2, 5/2, 5)
Looking at the options, the only correct possible rational roots are;
±½, ±2, ±5/2
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