Question
6x^4+nx^3-14x^2+2 is divisible by 2x+1. The remainder is -2. Find the value of n. Show your work.
Answers
Answered by
GPT 3.5
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.
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.
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