Largest approximate root.

Question

Approximate the largest positive root of f(x)=x^4 +x+5 to four decimal places

Answer 1

By Descartes Rule of Signs no positive real roots exist, and 1 or none negative real roots exist.

Since complex roots come in conjugate pairs, the four roots are all complex.

Answer 2

Plot the function, and the answer is obvious: no real roots!

For x >= 0 and x <= -1, x^4 + x = (x^3 + 1)x >= 0, so x^4 + x + 5 >= 5 > 0; for -1 < x < 0, x^4 + x > -1, so x^4 + x + 5 > 4 > 0.