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# Find the value of k for which the binomial (2x – 1) is a factor of 2x 3 + kx ​2 – 2x + 1

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Find the value of k for which the binomial (2x – 1) is a factor of 2x3 + kx2 – 2x + 1

Aug 4, 2019

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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 $$\mathbf{x=\dfrac12}$$

$$\begin{array}{|rcll|} \hline 2x^3 + kx^2 – 2x + 1 \quad &| \quad \mathbf{x=\dfrac12} \\ 2\left(\dfrac12\right)^3 + k\left(\dfrac12\right)^2 – 2\left(\dfrac12\right) + 1 &=& 0 \\ \dfrac{2}{8} + \dfrac{k}{4} – \dfrac{2}{2} + 1 &=& 0 \\ \dfrac{1}{4} + \dfrac{k}{4} – 1 + 1 &=& 0 \\ \dfrac{1}{4} + \dfrac{k}{4} &=& 0 \quad &| \quad \cdot 4\\ 1+k &=& 0 \quad &| \quad -1 \\ \mathbf{k} &=& \mathbf{-1} \\ \hline \end{array}$$

Aug 4, 2019