Multiple choice

The equation $x^ {4}+16x-12=0$ has

  1. all the roots real

  2. all the roots complex

  3. two roots real and two roots complex

  4. all the roots integers
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let f(x) = x^4 + 16x - 12. f'(x) = 4x^3 + 16. f'(x) = 0 at x = -cube root(4) approx -1.587. f(-1.587) = (-1.587)^4 + 16(-1.587) - 12 = 6.35 - 25.39 - 12 = -31.04. Since the local minimum is negative and the function goes to infinity at both ends, there are two real roots. The other two must be complex.

AI explanation

Let f of x equal x to the fourth power plus 16x minus 12. Since f of 0 is negative 12 and f of 1 equals 5, one real positive root exists between 0 and 1. Furthermore, as x approaches negative infinity, f of x is positive, and since f of negative 2 is 4 minus 32 minus 12, which is negative 40, a second real root exists between negative infinity and negative 2. Therefore, the polynomial has exactly two real roots; because complex roots must occur in conjugate pairs, the remaining two roots are complex, meaning the equation has two real roots and two complex roots.

Keep practicing — related questions