Given q (x) = 3x^3 - 4x^2 + 5x + k. a. Determine the value of k so that 3x - 7 is a factor of the polynomial q. b. What is the quotient when you divide the polynomial q by 3x - 7?

a) k = - 28

b) (x² + x + 4)

Step-by-step explanation:

Here, we are given the function q (x) = 3x³ - 4x² + 5x + k.

a) First, we have to find the value of k for which (3x - 7) will be a factor of q (x).

For this purpose, we will rearrange the function as follows:

q (x) = 3x³ - 4x² + 5x + k

= (3x³ - 7x² + 3x² - 7x + 12x - 28) + (k+28)

= [x² (3x-7) + x (3x-7) + 4 (3x-7) ] + (k+28)

= (3x-7) (x² + x + 4) + (k+28)

From the above expression it is clear that to make (3x-7) a factor of q (x), the extra term (k+28) has to be 0.

Therefore, (k+28) = 0, ⇒ k = -28 (Answer)

b) Now, if k = - 28, then q (x) becomes (3x-7) (x² + x + 4).

Hence, if we divide q (x) by (3x-7) then the quotient will be (x² + x + 4). (Answer)