Multiple choice

The equation $x(x+2)(x^{2}-x)=-1$, has

  1. All roots imaginary

  2. All roots negative

  3. Two roots real and two roots imaginary

  4. All roots real
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

x(x+2)(x^2-x) = -1. x(x+2)x(x-1) = -1. x^2(x+2)(x-1) = -1. x^2(x^2 + x - 2) = -1. x^4 + x^3 - 2x^2 + 1 = 0. Testing for real roots: let f(x) = x^4 + x^3 - 2x^2 + 1. f'(x) = 4x^3 + 3x^2 - 4x = x(4x^2 + 3x - 4). Roots of f'(x) are 0 and (-3 +/- sqrt(9 + 64))/8 = (-3 +/- 8.5)/8. The function values at these points are all positive, meaning the minimum value is positive. Thus, there are no real roots.

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