Question

For what value of k, – 4 is a zero of the polynomial equation \[{x^2} - x - \left( {2k + 2} \right)\]?

Answer 1

Hint: Since -4 is a zero of the equation substituting the value of x as -4 will make the equation’s value as zero, therefore do this to find the value of k.

Complete step-by-step answer:

As we know that the zero of the polynomial equation is that value which satisfies polynomial equation or we can say that when we put the variable of polynomial equation equal to that value then the value of the polynomial equation becomes zero.
So, as we are given that the polynomial equation is,
\[{x^2} - x - \left( {2k + 2} \right)\]
And – 4 is the zero of this polynomial equation.
So, when we put the value of x equal to – 4 in the above polynomial equation then the value of the polynomial equation must be equal to zero.
So, according to the question.
\[{\left( { - 4} \right)^2} - \left( { - 4} \right) - \left( {2k + 2} \right) = 0\]
On solving the above equation. We get,
$\Rightarrow$ 16 + 4 – 2k – 2 = 0
$\Rightarrow$ 18 – 2k = 0
Adding 2k to both the sides of the above equation. We get,
$\Rightarrow$ 18 = 2k
And dividing both sides of the above equation by 2. We get,
$\Rightarrow$ k = 9
Hence, for k = 9, – 4 will be the zero of the polynomial equation \[{x^2} - x - \left( {2k + 2} \right)\]

Note: Whenever we come up with this type of problem then we should remember that any polynomial should have a maximum number of zeros equal to the highest degree of that polynomial equation. And the highest degree of the polynomial equation is the highest power of the variable i.e. x of that equation. And a polynomial equation can have more than one zeros. So, if we are given that –4 is the zero of the polynomial equation then we put the value of x equal to –4 in the given equation and then equate that with zero to find the value of k.