Question

For what value of k, is the polynomial \[{{x}^{5}}-3{{x}^{4}}-k{{x}^{3}}+3k{{x}^{2}}+2kx+4\] exactly divisible by (x – 2)?

Answer 1

Hint: Assume the given polynomial as f (x). Now, consider (x – 2) as a factor of f (x) and equate (x – 2) with 0 to find the value of x. Considering this obtained value of x as a zero of the polynomial f (x), substitute f (x) = 0 at this value of x. Form a linear equation in ‘k’ and solve it to get its value.

Complete step-by-step solution
Here, we have been provided with a polynomial \[{{x}^{5}}-3{{x}^{4}}-k{{x}^{3}}+3k{{x}^{2}}+2kx+4\] and we have to find the value of k for which this polynomial will be divisible by (x – 2).
Now, let us assume f (x) = \[{{x}^{5}}-3{{x}^{4}}-k{{x}^{3}}+3k{{x}^{2}}+2kx+4\]. Here, (x – 2) divides f (x) completely. That means (x – 2) is a factor of f (x).
We know that if (x – a) is a factor of a polynomial f (x) then x = a is called the zero of the polynomial f (x). Here, x = a is obtained by substituting the given factor (x – a) equal to 0.
Now, let us come to the question. Substituting (x – 2) equal to 0, we get,
\[\begin{align}
  & \Rightarrow x-2=0 \\
 & \Rightarrow x=2 \\
\end{align}\]
Therefore, x = 2 is a zero of the polynomial.
We know that if x = a is a zero of the polynomial then f (a) = 0. Therefore, we have,
\[\Rightarrow f\left( 2 \right)=0\]
So, substituting x = 2 in the given polynomial f (x) = \[{{x}^{5}}-3{{x}^{4}}-k{{x}^{3}}+3k{{x}^{2}}+2kx+4\], we get,
\[\begin{align}
  & \Rightarrow {{2}^{5}}-3\times {{2}^{4}}-k\times {{2}^{3}}+2k\times 2+4=0 \\
 & \Rightarrow 32-48-8k+4k+4=0 \\
 & \Rightarrow -4k-12=0 \\
\end{align}\]
Taking the constant term to the R.H.S, we get,
\[\Rightarrow -4k=12\]
Dividing both sides with -4, we get,
\[\begin{align}
  & \Rightarrow k=\dfrac{-12}{4} \\
 & \Rightarrow k=-3 \\
\end{align}\]
Hence, the value of ‘k’ must be (-3) for which the polynomial f (x) must be exactly divisible with (x – 2).

Note: One may note that the process we have applied to solve the question is known as the factor theorem which is a special case of the polynomial remainder theorem. Remember that, do not try to divide the assumed polynomial f (x) with (x – 2) using the long division method. In this way also you will get the answer but it will take much time to solve the question. The two theorems must be remembered to solve the question easily.