Solution:

Given, x - 2 is a factor of polynomial x³ - 7x² + x - 16.

We need to find the value of k.

Using factor theorem:

If (x - a) is a factor of a polynomial, then the polynomial is completely divisible by (x - a). This means that if we substitute the value of 'a' in the polynomial and the result is zero, then (x - a) is a factor of the polynomial.

So, if (x - 2) is a factor of x³ - 7x² + x - 16, then we have:

(x - 2) divides x³ - 7x² + x - 16

Substituting x = 2, we get:

(2)³ - 7(2)² + (2) - 16 = 0
8 - 28 + 2 - 16 = 0
-34 = 0

This is not possible.

So, we can say that x - 2 is not a factor of x³ - 7x² + x - 16.

Using long division method:

We can use the long division method to divide x³ - 7x² + x - 16 by x - 2.

Step 1: Write the dividend and divisor.


x - 2 | x³ - 7x² + x - 16

Step 2: Divide the first term of the dividend by the divisor and write the result.



x - 2 | x³ - 7x² + x - 16

Step 3: Multiply the divisor by the quotient obtained in step 2 and write the result below the dividend.



x - 2 | x³ - 7x² + x - 16


Step 4: Subtract the result obtained in step 3 from the dividend and write the result below.



x - 2 | x³ - 7x² + x - 16




Step 5: Repeat steps 2 to 4 with the new dividend.



x - 2 | x³ - 7x² + x - 16

The value of k is -2