Which of the following polynomials with integer coefficients has $\sin\left(10^{\circ}\right)$ as a root?

$\text{(A) }4x^3-4x+1\qquad\text{(B) }6x^3-8x^2+1\qquad\text{(C) }4x^3+4x-1\qquad\text{(D) }8x^3+6x-1\qquad\text{(E) }8x^3-6x+1$
To determine which polynomial has $\sin(10^\circ)$ as a root, we can use the triple angle identity for sine, which states:
\[
\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)
\]
Let $\theta = 10^\circ$. Then, $\sin(30^\circ) = \frac{1}{2}$, and we can set up the equation:
\[
\sin(30^\circ) = 3\sin(10^\circ) - 4\sin^3(10^\circ)
\]
Let $a = \sin(10^\circ)$. Then, the equation becomes:
\[
\frac{1}{2} = 3a - 4a^3
\]
To clear the fraction, multiply both sides by 2:
\[
1 = 6a - 8a^3
\]
Rearrange the equation to form a polynomial:
\[
8a^3 - 6a + 1 = 0
\]
Thus, the polynomial with integer coefficients that has $\sin(10^\circ)$ as a root is:
\[
8x^3 - 6x + 1
\]

The final answer is $\boxed{8x^3 - 6x + 1}$