Find the value of k for which the binomial (2x – 1) is a factor of 2x3 + kx2 – 2x + 1
+26367 Find the value of \(k\) for which the binomial \((2x – 1)\) is a factor of \(2x^3 + kx^2 – 2x + 1\)
\(\begin{array}{|rcll|} \hline (2x – 1)&=& 2\left(x – \mathbf{\dfrac12} \right) \\ \hline \end{array}\)
The root is at \(\mathbf{x=\dfrac12}\)
\(\begin{array}{|rcll|} \hline 2x^3 + kx^2 – 2x + 1 \quad &| \quad \mathbf{x=\dfrac12} \\ 2\left(\dfrac12\right)^3 + k\left(\dfrac12\right)^2 – 2\left(\dfrac12\right) + 1 &=& 0 \\ \dfrac{2}{8} + \dfrac{k}{4} – \dfrac{2}{2} + 1 &=& 0 \\ \dfrac{1}{4} + \dfrac{k}{4} – 1 + 1 &=& 0 \\ \dfrac{1}{4} + \dfrac{k}{4} &=& 0 \quad &| \quad \cdot 4\\ 1+k &=& 0 \quad &| \quad -1 \\ \mathbf{k} &=& \mathbf{-1} \\ \hline \end{array}\)
