# Find k such that the Polynomial P(x)=-2x+kx^2+2x+4 is divisible by x-2.

Find k such that the polynomial $P\left(x\right)=-2{x}^{3}+k{x}^{2}+2x+4$ is divisible by x-2.
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$P\left(x\right)=-2{x}^{3}+k{x}^{2}+2x+4$
As $\left(x-1\right)$ is a factor of polynomial $p\left(x\right)=-2{x}^{3}+kx+2x+4$
So $p\left(2\right)=0\phantom{\rule{0ex}{0ex}}⇒-2\left(2{\right)}^{3}+k\left(2{\right)}^{2}+2\left(2\right)+4=0\phantom{\rule{0ex}{0ex}}⇒-2\left(8\right)+k\left(4\right)+4+4=0\phantom{\rule{0ex}{0ex}}⇒16+4k+8=0\phantom{\rule{0ex}{0ex}}⇒-8+4k=0\phantom{\rule{0ex}{0ex}}⇒4k=8\phantom{\rule{0ex}{0ex}}k=\frac{8}{4}\phantom{\rule{0ex}{0ex}}k=2$