Let $p(x)$ be a quadratic polynomial such that $p(1) = p(-1) = 0$. What is the coefficient of $x$ in $p(x)$?

Let $p(x) = ax^2 + bx + c$ be the quadratic polynomial, where $a \neq 0$.

Given $p(1) = 0$, we have $a + b + c = 0$.

Given $p(-1) = 0$, we have $a - b + c = 0$.

Subtracting the second equation from the first gives $2b = 0$, which implies $b = 0$.

Alternatively, since the roots of the quadratic polynomial are $1$ and $-1$, the sum of the roots is $1 + (-1) = 0$.

Using the relation between roots and coefficients, the sum of the roots is $-\dfrac{b}{a} = 0$, which gives $b = 0$.

Therefore, the coefficient of $x$ in $p(x)$ is $0$.

Answer: $0$