Answer:
a) k = -28
b) (x² +x +4)
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)