find anumber k satisfying the given condition. 3x-2 is afactor of 6xcube-4xsq+kx-k

To find a number "k" that satisfies the condition, we need to use the concept of factorization and equate the given polynomial expression to zero.

The given condition is that 3x - 2 is a factor of the polynomial expression 6x^3 - 4x^2 + kx - k.

When a linear expression is a factor of a polynomial, it means that when we substitute the root of the linear expression into the polynomial, the result is zero. In this case, substituting 3x - 2 into the polynomial should give us zero.

So, we can set up the equation:

6x^3 - 4x^2 + kx - k = (3x - 2)(...)

To determine the rest of the factorization, we can divide the polynomial (6x^3 - 4x^2 + kx - k) by (3x - 2) using either long division or synthetic division.

Let's perform the long division:

2x^2 + x + 1
___________________
3x - 2 | 6x^3 - 4x^2 + kx - k
- (6x^3 - 4x^2)
_______________
7x^2 + kx
- (7x^2 - 4x)
_______________
5x + k

We can stop here since there is no remainder.

Therefore, the expression can be fully factored as:
6x^3 - 4x^2 + kx - k = (3x - 2)(2x^2 + x + 1)

Now, we need to find the value of "k". Since (3x - 2) is a factor of the polynomial, it means that substituting x = 2/3 into the polynomial should make it zero.

Let's substitute x = 2/3 into the polynomial:
6(2/3)^3 - 4(2/3)^2 + k(2/3) - k = 0

Evaluating the expression:
8/9 - 16/9 + 2k/3 - k = 0

Combining like terms:
(2k/3 - k) = (16/9 - 8/9)

Simplifying:
(2k/3 - k) = 8/9

Multiplying both sides by 9 to eliminate the denominator:
3(2k) - 9k = 8

Simplifying further:
6k - 9k = 8
-3k = 8
k = -8/3

Therefore, the number "k" that satisfies the given condition is k = -8/3.