6x^4+nx^3-14x^2+2 is divisible by 2x+1. The remainder is -2. Find the value of n. Show your work.

Answer 1

To find the value of n, we will use the remainder theorem for polynomial functions.

Given that 6x^4 + nx^3 - 14x^2 + 2 is divisible by 2x + 1, we know that when we substitute x = -1/2 into the polynomial, the result should be equal to the remainder, which is -2.

Substitute x = -1/2 into the polynomial:

6(-1/2)^4 + n(-1/2)^3 - 14(-1/2)^2 + 2 = -2

6(1/16) - n(1/8) - 14(1/4) + 2 = -2

3/8 - n/8 - 14/4 + 2 = -2

3/8 - n/8 - 14/4 + 2 = -2

3/8 - n/8 - 7/2 + 2 = -2

Convert fractions to have the same denominators and solve for n:

3/8 - n/8 - 28/8 + 16/8 = -16/8

-25/8 - n/8 = -16/8

-n/8 = -16/8 + 25/8

-n/8 = 9/8

n = -9

Therefore, the value of n is -9.