Show that: 2x - 3 is a factor of x + 2x^3 - 9x^2 + 12. - Vidyadip

Question

Show that: $2x - 3$ is a factor of $x + 2x^3 - 9x^2 + 12.$

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Answer

Let $p(x) = x + 2x^3 - 9x^2 + 12, g(x) = 2x - 3$
$g(x) = 2x - 3 = 0$ gives $\text{x}=\frac{3}{2}$
$g(x)$ will be a factor of$ p(x)$ if $\text{p}\Big(\frac{3}{2}\Big)=0$ (Factor theorem)
Now, $\text{p}\Big(\frac{3}{2}\Big)=\frac{3}{2}+2\Big(\frac{3}{2}\Big)^3-9\Big(\frac{3}{2}\Big)^2+12$
$=\frac{3}{2}+2\Big(\frac{27}{8}\Big)-9\Big(\frac{9}{4}\Big)+12$
$=\frac{3}{2}+\frac{27}{4}-\frac{81}{4}+12$
$=\frac{6+27-81+48}{4}=\frac{0}{4}=0$
Since, $\text{p}\Big(\frac{3}{2}\Big)=0,$ So $g(x)$ is a factor of $p(x).$

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