2023-02-25

How to use the rational root theorem to list all possible rational roots?

### Answer & Explanation

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$f\left(x\right)={a}_{0}{x}^{n}+{a}_{1}{x}^{n-1}+...+{a}_{n-1}x+{a}_{n}$
Any rational roots of f(x) = 0 may be expressed in lowest terms as $\frac{p}{q}$, where p, $p,q\in ℤ$, $q\ne 0$, p is a divisor of ${a}_{n}$, and q is a divisor of ${a}_{0}$. To discover all the distinct fractions $\frac{p}{q}$ that arise, list all the possible integer factors of ${a}_{0}$ and ${a}_{n}$, then all the potential rational roots. Consider the case when $f\left(x\right)=6{x}^{3}-12{x}^{2}+5x+10$
${a}_{n}=10$ has factors +-1, +-2, +-5 and +-10. (possible values of p); ${a}_{0}=6$ has factors +-1, +-2, +-3 and +-6. (possible values of q)
Possible fractions:
+-1/6, +-1/3, +-1/2, +-2/3, +-5/6, +-1, +-5/3, +-2, +-5/2, +-10/3, +-5, +-10

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