Question

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

Answer 1

Hint: The zero of a polynomial is defined as the value of the variable when the polynomial vanishes. Substitute the value of x = – 4 and equate it to zero to find the value of k as it is the zero of the given polynomial, So it will return the value of the polynomial.

Complete step-by-step solution -
A polynomial is defined as an expression which contains two or more algebraic terms. It involves constants, variables and exponents. For example, \[3x + 5\] is a polynomial.
A zero of a polynomial is the value of the variable for which the value of the polynomial becomes zero. It is also called the root or the solution of the equation P(x) = 0, where P(x) is the polynomial.
In our question, we are given the polynomial \[{x^2} - x(2k + 2)\]. We have:
\[P(x) = {x^2} - x(2k + 2)............(1)\]
We need to find the value of k and it is given that – 4 is a zero of this polynomial. By the definition of the zero of the polynomial, we have:
\[P( - 4) = 0\]
Substituting equation (1) in the above equation, we have as follows:
\[{( - 4)^2} - ( - 4)(2k + 2) = 0\]
Simplifying the above equation, we get:
\[16 + 4(2k + 2) = 0\]
Expanding the terms in the bracket, we have:
\[16 + 8k + 8 = 0\]
Adding the numbers 16 and 8, we have as follows:
\[24 + 8k = 0\]
Solving for k, we have:
\[k = - \dfrac{{24}}{8}\]
\[k = - 3\]
Hence, the value of k is – 3.

Note: The possibility of making a mistake when evaluating the term – (– 4) is high. Its value is + 4. If we write it as – 4, it is incorrect and we will get a wrong value for k.