Question

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?

Answer 1

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)