Chapter 5 - Section 5.5 - The Real Zeros of a Polynomial Function - 5.5 Assess Your Understanding - Page 387: 15

$R=0$ Since $R=0$, $x+3$ is a factor.

$f\left( x\right) =3x^{6}+82x^{3}+27=\left( x+3\right) q\left( x\right) +R$ $\Rightarrow f\left( -3\right) =3\times \left( -3\right) ^{6}+82\left( -3\right) ^{3}+27=0=R$ Since $R=0$, $x+3$ is a factor.