Question

Find the value of k if ( x – 2 ) is a factor of $p(x) = {x^2} + kx + 2k$

Answer 1

Hint: We are given that ( x – 2) is a factor of p(x). then by factor theorem when ( x – a ) is a factor of p(x) then p(a) = 0. Hence by substituting 2 in the place of x and equating it to 0 we get the value of k

Complete step-by-step answer:
We are given a polynomial $p(x) = {x^2} + kx + 2k$
And ( x – 2) is the factor of p(x)
Then by using factor theorem when ( x – a ) is a factor of p(x) then p(a) = 0
From the given factor we get the value of x to be
$
   \Rightarrow x - 2 = 0 \\
   \Rightarrow x = 2 \\
$
Now as it is a factor we have p(2) to be 0
$
   \Rightarrow p(2) = {2^2} + 2k + k \\
   \Rightarrow 0 = 4 + 3k \\
   \Rightarrow - 4 = 3k \\
   \Rightarrow k = \dfrac{{ - 4}}{3} \\
$
Hence we obtain the value of k.

Note: This can also be done using division algorithm
As ( x – 2 ) is a factor then the remainder when p(x) is divided by ( x – 2 ) is zero
$\begin{gathered}
   \Rightarrow x - 2\mathop{\left){\vphantom{1\begin{gathered}
  {x^2} + kx + k \\
  \underline {{x^2} - 2x} \\
  {\text{ }}(k + 2)x + k \\
  {\text{ }}\underline {(k + 2)x - 2(k + 2)} \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  {x^2} + kx + k \\
  \underline {{x^2} - 2x} \\
  {\text{ }}(k + 2)x + k \\
  {\text{ }}\underline {(k + 2)x - 2(k + 2)} \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, {x + (k + 2)}} \\
  {\text{ }}k + 2(k + 2) \\
\end{gathered} $
 As it is a factor the remainder must be zero
$
   \Rightarrow k + 2(k + 2) = 0 \\
   \Rightarrow k + 2k + 4 = 0 \\
   \Rightarrow 3k + 4 = 0 \\
   \Rightarrow k = \dfrac{{ - 4}}{3} \\
$