Answer
There is a root in the specified interval.
Work Step by Step
$f(x) = x^{4} + x - 3$
Use the numbers of the given interval: $(1,2)$
$f(1) = 1^{4} + 1 - 3 = -1$
$f(2) = 2^{4} + 2 - 3 = 15$
From this we know that $f(x)$ is a continuous function in the given interval.
By the definition of the Intermediate Value Theorem $f(x)$ will take all the values between $-1$ and $15$. So there is at least one value of $x$ for which $f(x)$ is zero. So there is at least one root of $f(x) = 0$ in the interval $(1,2)$.
