Question

List all the potential rational zeros of $f\left(x\right)=12x^8-x^7+6x^4-x^3+x-3$.

Short answer

$\frac{p}{q}=\pm 1,\pm 3,\pm \frac{1}{2},\pm \frac{3}{2},\pm \frac{1}{3},\pm \frac{1}{4},\pm \frac{3}{4},\pm \frac{1}{6},\pm \frac{1}{12}$

Step-by-step solution

Given information

The given polynomial function is$f\left(x\right)=12x^8-x^7+6x^4-x^3+x-3$

Determine the all possible zero of the given function.

List all the possible factors $p$of $a_0$.

$p=\pm 1,\pm 3$

List all the possible factors $q$of $a_8$.

$q=\pm 1,\pm 2,\pm 3,\pm 4\pm 6,\pm 12$

Identify all possible rational zeros $\frac{p}{q}$.

$\frac{p}{q}=\pm 1,\pm 3,\pm \frac{1}{2},\pm \frac{3}{2},\pm \frac{1}{3},\pm \frac{1}{4},\pm \frac{3}{4},\pm \frac{1}{6},\pm \frac{1}{12}$

Write the conclusion

The possible real zeros are$\pm 1,\pm 3,\pm \frac{1}{2},\pm \frac{3}{2},\pm \frac{1}{3},\pm \frac{1}{4},\pm \frac{3}{4},\pm \frac{1}{6},\pm \frac{1}{12}$.