Examine whether $2 x+3$ is a factor of $2 x^{3}+21 x^{2}+67 x+60$ or not.
Examine whether $2 x+3$ is a factor of $2 x^{3}+21 x^{2}+67 x+60$ or not.
The zero of $2 x+3$ is $\left(-\frac{3}{2}\right)$
Now, $p(x)=2 x^{3}+21 x^{2}+67 x+60$
$\therefore p\left(-\frac{3}{2}\right)=2\left(-\frac{3}{2}\right)^{3}+21\left(-\frac{3}{2}\right)^{2}+67\left(-\frac{3}{2}\right)+60$
$=2\left(-\frac{27}{8}\right)+21\left(\frac{9}{4}\right)-\frac{201}{2}+60$
$=-\frac{27}{4}+\frac{189}{4}-\frac{201}{2}+60$
$=\frac{-27+189-402+240}{4}$
$=\frac{-429+429}{4}=\frac{0}{4}$
$\therefore p\left(-\frac{3}{2}\right)=0$
So, by the factor theorem, $2 x+3$ is a factor of $2 x^{3}+21 x^{2}+67 x+60$.
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