Question

2. 6x
+ nx - 14x + 2 is divisible by 2x + 1. The remainder is - 2. Find the value of n. Justify your answer by showing your work.

Answer 1

To find the value of n, we first need to perform polynomial long division with the given expression and the divisor 2x + 1.

6x + nx - 14x + 2 divided by 2x + 1

First, we divide the first term of the dividend by the first term of the divisor:
6x / 2x = 3

Next, multiply the divisor by 3:
3(2x + 1) = 6x + 3

Subtract this from the original dividend:
(6x + nx - 14x + 2) - (6x + 3) = (nx - 14x - 6x) + (2 - 3) = (nx - 20x) - 1

Now, we repeat the process with the new dividend (nx - 20x - 1) divided by 2x + 1:

(n - 20)x - 1 divided by 2x + 1

Divide the first term of the new dividend by the first term of the divisor:
(n - 20)x / 2x = (n - 20) / 2

Since the remainder is -2, the division should result in a remainder of -2. Therefore, our divisor must contain (n - 20) / 2 term without the variable 'x'.

(n - 20) / 2 = -2
n - 20 = -4
n = 16

Therefore, the value of n is 16.