Calculus: Early Transcendentals, Chapter 4, 4.2, Section 4.2, Problem 18

Denote f(x)=x^3+e^x. f is continuous everywhere.
When x tends to +oo , f(x) tends to +oo , when x tends to -oo, f(x) tends to -oo. In another words, there are x_1 such that f(x_1) lt 0 (for example, x_1=-1 ) and x_2 such that f(x_2)gt0 (for example, x_2=0 ).
By the Intermediate Value Theorem there is at least one c such that f(c)=0.
Let's prove that such c is unique. Really, f(x) is increases strictly monotonically as a sum of two such functions. If we want, we can use the derivative, f'(x)=3x^2 + e^x gt 0. Therefore f takes any value only once, QED.