Introduction to Polynomials and Zeros

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. Understanding how to find the zeros of a polynomial is crucial in algebra, and this article will guide you through the steps to find the value of k for which 2 is a zero of the specific polynomial given.

Defining the Polynomial and the Problem

The polynomial in question is given by P(x) x^3 - 3x^2 4x 2k. We are tasked with finding the value of k such that 2 is a zero of this polynomial.

Solving for k

To find k for which 2 is a zero, we substitute x 2 into the polynomial and set it equal to zero:

[P(2) 2^3 - 3(2)^2 4(2) 2k 0]

Now, we will simplify each term step by step:

Calculate [2^3 8].

Calculate [3(2)^2 3(4) 12].

Calculate [4(2) 8].

Substituting these values back into the equation:

[8 - 12 8 2k 0]

Simplify the left-hand side: [8 - 12 8 4].

The equation now simplifies to: [4 2k 0].

Solving for k, we get: [2k -4] and [k -2].

Thus, the value of k is [boxed{-2}].

Conclusion

In this article, we went through the step-by-step process of finding the value of k for which the polynomial x^3 - 3x^2 4x 2k has 2 as a zero. By substituting x 2 and setting the polynomial equal to zero, we solved for k and determined that k -2. Understanding how to solve similar problems can help in more advanced mathematical and algebraic operations.