Is there a quick way of determining where a polynomial is positive/negative without actually plugging values?
Say you have a polynomial
$$1) f(x)=(x+a)(x+b)$$
or
$$2) f(x)=(x-a)(-x+b)(x-c)$$
How can one quickly know whether function $f(x)$ is positive or negative without plugging in random interval values between the roots?
On
Sure. If you have those factorizations, then it's not so bad. You want to look at the sign of each individual term.
In the case of $(1)$, we know that $f(x)$ will be positive when $(x+a)$ and $(x+b)$ are the same sign, and negative when they are different. When is $x + a < 0$? Well, when $x < -a$. When is $x + b < 0$? Similarly, when $x < -b$. Without loss of generality, assume $a > b$.
Then when $x < -a$, we also have $x < -b$. So we would have $f(x) = (x+a)(x+b) = (-)(-) = (+)$.
How about when $-a < x < -b$? Then we have $f(x) = (-)(+) = (-)$.
And when $-a < -b < x$? Then we have $f(x) = (+)(+) = (+)$.
Of course, when you have either $x = -a$ or $x = -b$, then $f(x) = 0$. And if $a = b$, $f(x) = (x+a)^{2}$, so it's always nonnegative, and zero only when $x = -a$.
Using this kind of analysis, can you try your second example?
Looking at $(x+a)(x+b)$, if $x$ is "large" then the polynomial is positive. Now imagine $x$ decreasing. Provided $a\neq b$, once $x$ crosses the larger of $-a, -b$, $f(x)$ will change sign. Once $x$ crosses the other one, it will change sign again.
This strategy needs to be modified if any of the roots are repeated. For example, $f(x)=(x-2)^5(x-3)^4$. This is positive for "large" $x$. Crossing below $3$ the sign does not change, because $4$ is even, so $f(x)$ is positive in $(2,3)$. Crossing below $2$ the sign does change, because $5$ is odd, so $f(x)$ is negative on $(-\infty, 2)$.