Determine the intervals on which f(x) is positive and on which f(x) is negative. f(x)=x^4–3x^3+8x^2–10x+8?

f(x) = x^4 – 3x^3 + 8x^2 – 10x + 8

(x²-2x+2) (x²-x+4) = 0

x²-2x+2 = 0

x = (2 ± √4-8)/2

= 1 ± i

(x²-x+4) = 0

x = (1 ± √1-4)2

= (1 ± i√3)/2

A little algebra (factoring) shows that ur eqs can be written as

(x²-2x+2)*(x²-x+4).

Each of these have no real roots.

So either your f(x) is all positive or negative.

Since f(0) = 8 > 0 then its all positive.

BTW, there are other ways to attain this conclusion.

A graph of the function shows that all values of x yield positive values of f(x).

(-∞,∞) is the interval over which f(x) is positive.