Question

Determine the intervals on which f(x) is postitive or negative f(x)=x^5+x^4-3x^3-3x^2-9x-9

Answer 1

Answer:

• positive: (-2.2032, -1) ∪ (2.2032, ∞)
• negative: (-∞, -2.2032) ∪ (-1, 2.2032)

Step-by-step explanation:

The given function is an odd-degree polynomial with a positive leading coefficient. This means extreme values of x will give function values with the same sign. The function will be negative to the left of its leftmost zero, and will be positive to the right of its rightmost zero. It will change sign at each zero crossing.

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graph

A graph shows there are three real zeros: 1 rational, and 2 irrational. The remaining two zeros are complex, so have no effect on the sign of the function.

zeros

The rational zero at x = -1 can be factored out using synthetic division (2nd attachment).

The factorization is then ...

  f(x) = (x +1)(x^4 -3x^2 -9)

roots of the quartic

The quartic factor is a quadratic in x^2, so can be written in a sort of "vertex form" that can help us find its linear factors.

  x^4 -3x^2 -9 = (x^2 -1.5)^2 -11.25  ⇒  x^2 = 1.5 ±√11.25

For the positive value on the right side of this equation, the roots are real. For the negative value, they are imaginary.

  real roots: x = ±√(1.5 +√11.25) ≈ ±2.2032027

  complex roots:  x = ±√(1.5 -√11.25) ≈ ±1.3616541i

sign changes

So, the sign of f(x) is ...

• negative left of x ≈ -2.2032
• positive for -2.2032 < x < -1
• negative for -1 < x < 2.2032
• positive right of x = 2.2032