Respuesta :
We want to calculate the following division
[tex]\frac{x^4-2x^2+kx+6}{x-2}[/tex]To efetuate this divison, we start by dividing the leading term of the dividend by the leading term of the divisor:
[tex]\frac{x^4}{x}=x^3[/tex]Then, we multiply this result by the divisor:
[tex]x^3(x-2)=x^4-2x^3[/tex]Then, subtract the dividend from the obtained result:
[tex](x^4-2x^2+kx+6)-(x^4-2x^3)=2x^3-2x^2+kx+6[/tex]Then, we can rewrite our division as:
[tex]\frac{x^{4}-2x^{2}+kx+6}{x-2}=x^3+\frac{2x^3-2x^2+kx+6}{x-2}[/tex]The remainder of this first division still have a polynomial on the dividend, therefore, we can iterate the previous process until we find the final result of this division.
[tex]\begin{gathered} \frac{x^{4}-2x^{2}+kx+6}{x-2}= x^{3}+\frac{2x^{3}-2x^{2}+kx+6}{x-2} \\ =x^3+2x^2+\frac{2x^2+kx+6}{x-2} \\ =x^3+2x^2+2x+\frac{x(k+4)+6}{x-2} \\ =x^3+2x^2+2x+k+4+\frac{2k+14}{x-2} \end{gathered}[/tex]And this is the final result of the division.
[tex]\frac{x^4-2x^2+kx+6}{x-2}=x^3+2x^2+2x+k+4+\frac{2k+14}{x-2}[/tex]We want to find the value of k that will make the remainder equal to
The remainder of this division is
[tex]\frac{2k+14}{x-2}[/tex]Then, we just need to solve for k the following equation
[tex]\begin{gathered} \frac{2k+14}{x-2}=0\implies2k+14=0 \\ 2k=-14 \\ k=-\frac{14}{2} \\ k=-7 \end{gathered}[/tex]The value of k must be - 7.
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