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2x - 3 is a factor of x + 2x3 - 9x2 + 12.
2x - 3 is a factor of x + 2x3 - 9x2 + 12.
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Answer
Let p(x) = x + 2x3 - 9x2 + 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).
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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