How many real roots does the polynomial f(x)=x^7+5x+12 have?

Question

How many real roots does the polynomial f(x)=x^7+5x+12 have?

Answer 1

f(-x) = -x² - 5x + 12

By Descartes Rule of signs, one negative real.

Answer 2

You cannot analytically solve this polynomial. It has to be done numerically. I put the function into Excel and evaluated it from -10 to 10 in intervals of 1. At x = -10, the function evaluates as a very large negative number. As x increases the numbers become smaller negative numbers and then have a zero crossing between x = -2 and x = -1. For all x > -1, the function becomes larger and stays positive. So, numerically, the function has one real root and 6 complex roots.