Question

The value of $k$ for which for polynomial $2{x^3} - {x^2} + 3x - k$ is divisible by $\left( {x - 1} \right)$ is
A) 1
B) 2
C) 3
D) 4

Answer 1

Hint: Here, we are given a cubic polynomial and we are given one of the factors $\left( {x - 1} \right)$ of this polynomial. That means when we divide this polynomial with $\left( {x - 1} \right)$, we get the remainder 0. Therefore, when we substitute x=1 in the polynomial, we get answer 0. Solving the obtained equation, we will get the value of k.

Complete step by step solution:
In this question, we are given a cubic polynomial and we are given its one of the factors and we need to find the constant k.
The given polynomial is : $2{x^3} - {x^2} + 3x - k$- - - - - - - - - - - - - (1)
Now, we are given that this polynomial is divisible by (x-1). That means when we divide$2{x^3} - {x^2} + 3x - k$ with (x-1), we get the remainder as 0. So that means when we put $x = 1$ in the given polynomial, we will get remainder 0. Therefore, we can write
$ \Rightarrow 2{x^3} - {x^2} + 3x - k = 0$- - - - - - - - (2)
Now, put $x = 1$ in equation (2), we get
$
   \Rightarrow 2{\left( 1 \right)^3} - {\left( 1 \right)^2} + 3\left( 1 \right) - k = 0 \\
   \Rightarrow 2 - 1 + 3 - k = 0 \\
   \Rightarrow k = 1 + 3 \\
   \Rightarrow k = 4 \\
 $
Hence, the value of $k$ for the polynomial $2{x^3} - {x^2} + 3x - k = 0$ is 4. Therefore, option (D) is the correct answer.

Note:
Here, we can cross check our answer by putting the value of k equal to 4 in the given polynomial and as we are given that (x-1) is one of the factors of this polynomial, after substituting the value of x as 1, we should get answer as 0. Therefore,
$
   \Rightarrow 2{\left( 1 \right)^3} - {\left( 1 \right)^2} + 3\left( 1 \right) - 4 = 0 \\
   \Rightarrow 2 - 1 + 3 - 4 = 0 \\
   \Rightarrow 1 + 3 - 4 = 0 \\
   \Rightarrow 4 - 4 = 0 \\
   \Rightarrow 0 = 0 \\
 $
Hence, our answer is correct.