Solving f(x)+x^4-6x^3+16x^2-26x+15

Question

How do I find that all zeros of this problem? What is the behavior? What is the function positive in?

What is the function is negative in? Show ste bpy step .

Answer 1

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Answer 2

I think you mean f(x) = x⁴ – 6x³ + 16x² – 26x + 15. By guessing, we discover, f(1) = 0.

So we need to do polynomial division f(x)/(x – 1) = x³ – 5x² + 11x – 15.

The domain and range of f(x)/(x – 1) are clearly both (–∞,∞) so it must have at least one real root, which after numerically plotting f(x)/(x – 1) vs x can be seen to be x → 3.

So we need to do polynomial division f(x)/(x – 1)(x – 3) = x² – 2x + 5 = (x – 1)² + 4 > 0, by completing the square.

So f(x) = (x – 1)(x – 3)((x – 1)² + 4) goes to ∞ as x → ±∞, has zeros at x → 1 and x → 3, is positive on (–∞,1) and on (3,∞), and negative on (1,3).