When using the rational root theorem, which of the following is a possible root of the polynomial function below?
[tex]F(x)=3 x^3-x^2+4 x+5[/tex]
A. -7
B. [tex]$-\frac{5}{3}$[/tex]
C. [tex]$\frac{4}{3}$[/tex]
D. 6
Answer (1)
The rational root theorem helps to identify potential rational roots of a polynomial. By listing the factors of the constant term and the leading coefficient, we form possible rational roots as fractions. Checking the given options against this list, we find that − 3 5 is a possible root. Therefore, the answer is − 3 5 .
Explanation
Understanding the Rational Root Theorem We are asked to find a possible root of the polynomial function F ( x ) = 3 x 3 − x 2 + 4 x + 5 using the rational root theorem. The rational root theorem states that if a polynomial has integer coefficients, then every rational root of the polynomial has the form q p where p is a factor of the constant term and q is a factor of the leading coefficient.
Listing Possible Rational Roots The constant term of the polynomial is 5, and its factors are ± 1 , ± 5 . The leading coefficient is 3, and its factors are ± 1 , ± 3 . Therefore, the possible rational roots are ± 1 , ± 5 , ± 3 1 , ± 3 5 .
Checking the Options Now we check the given options to see which one is in our list of possible rational roots:
A. -7 is not in the list. B. − 3 5 is in the list. C. 3 4 is not in the list. D. 6 is not in the list.
Conclusion Therefore, the only possible rational root among the given options is − 3 5 .
Examples
The rational root theorem is useful in various real-world scenarios, such as determining the possible dimensions of a rectangular box with a specific volume and integer side lengths. For instance, if the volume of a box is given by a polynomial equation, the rational root theorem can help identify potential integer or rational solutions for the side lengths, aiding in the design and optimization of physical structures.
