Question

Find the value of$k$, if $x–1$is a factor of $P\left(x\right)$ in the following case: $P\left(x\right)=k{x}^2-\sqrt{2}x+1$

Answer 1

Step 1: State the given data and the factor theorem

It is given that $P\left(x\right)=k{x}^2-\sqrt{2}x+1$ is the dividend.

And, $g\left(x\right)=x–1$ is the divisor.

We know, a polynomial $P\left(x\right)$ divided by $g\left(x\right)$ results in the quotient $Q\left(x\right)$ with zero remainders if and only if $Q\left(x\right)$ is a factor of $P\left(x\right)$.

Let us put divisor as equal to zero.

$\begin{array}{cr}& g\left(x\right)=0\\ \Rightarrow & x–1=0\\ \Rightarrow & x=1\end{array}$

Step 2: Apply factor theorem

If $x–1$ is a factor of $P\left(x\right)$, then by the factor theorem,

$\Rightarrow$ $P\left(1\right)=0$

$\Rightarrow$ $k{\left(1\right)}^2-\sqrt{2}\left(1\right)+1=0$

$\Rightarrow$ $k-\sqrt{2}+1=0$

$\Rightarrow$ $k=\sqrt{2}-1$

Hence, the required value of $k$ is $\sqrt{2}-1$.