Use the Rational Roots Test to Find All Possible Roots f(x)=12x^4+127x^3+234x^2-540x+216

$f (x) = 12 x^{4} + 127 x^{3} + 234 x^{2} - 540 x + 216$

Step 1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 9, \pm 12, \pm 18, \pm 24, \pm 27, \pm 36, \pm 54, \pm 72, \pm 108, \pm 216$

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

Step 2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{12}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm 4, \pm \frac{4}{3}, \pm 6, \pm 8, \pm \frac{8}{3}, \pm 9, \pm \frac{9}{2}, \pm \frac{9}{4}, \pm 12, \pm 18, \pm 24, \pm 27, \pm \frac{27}{2}, \pm \frac{27}{4}, \pm 36, \pm 54, \pm 72, \pm 108, \pm 216$

Use the Rational Roots Test to Find All Possible Roots f(x)=12x^4+127x^3+234x^2-540x+216

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