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Question: If \(\left( {x - 1} \right)\) is a polynomial of \(3{x^3} - 2{x^2} + kx - 6\) then what is the value...

If (x−1)\left( {x - 1} \right) is a polynomial of 3x3−2x2+kx−63{x^3} - 2{x^2} + kx - 6 then what is the value of x?

Explanation

Solution

In the question, we are given that the (x−1)\left( {x - 1} \right) is a polynomial of the equation given. This means that if we put this value into the equation will give the value zero. So, put x=1x = 1 into the equation and equate it to zero. Solve the above steps. Hence, we will find the answer.

Complete step by step solution:
In the above question, we are given that the (x−1)\left( {x - 1} \right) is a polynomial of 3x3−2x2+kx−63{x^3} - 2{x^2} + kx - 6 which means that if we equate the (x−1)\left( {x - 1} \right) to zero which gives
x−1=0 or x=1  x - 1 = 0 \\\ or \\\ x = 1 \\\
And then by putting this value to the polynomial then it will give us the value zero. Hence, putting the value into the polynomial
3(1)3−2(1)2+k(1)−6=03{\left( 1 \right)^3} - 2{\left( 1 \right)^2} + k\left( 1 \right) - 6 = 0
Solving the above equation to find the value of k,
3−2+k−6=0 ⇒k=5  3 - 2 + k - 6 = 0 \\\ \Rightarrow k = 5 \\\
Hence, the solution for the question is k=5k = 5

Note: Be careful while equating to zero. Because some students take x+1=0x + 1 = 0 to be x=1x = 1 which is totally wrong. These silly mistakes could lead to make the whole question wrong. While taking the values to the left-hand side the sign will change and vice-versa.