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.
+ nx - 14x + 2 is divisible by 2x + 1. The remainder is - 2. Find the value of n. Justify your answer by showing your work.
Answers
GPT 3.5
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.
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.