<s>[INST] If $4x^{10}-x^9-3x^8+5x^7+kx^6+2x^5-x^3+kx^2+5x-5 $ when divided by $(x+1)$ gives a remainder of -14, then the value of k equals? If $4x^{10}-x^9-3x^8+5x^7+kx^6+2x^5-x^3+kx^2+5x-5 $ when divided by $(x+1)$ gives a remainder of -14, then the value of k equals?
I got this and similar type of question in a book and I don't really know how to exactly solve it. Any help will be appreciated.
 [/INST] Let $f(x)=4x^{10}-x^9-3x^8+5x^7+kx^6+2x^5-x^3+kx^2+5x-5 $. Then $f(-1)=-14$ since the remainder when we divide by $x+1$ is $-14$.
So, $4+1-3-5+k-2+1+k-5-5=-14$
Thus, $2k=4$and $k=2$
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