In this case it's just a matter of plugging in values.

A polynomial is an equation with a lot of terms.

Roots are values that when plugged into the equation make the value of the equation 0.

If x were 1,2, or,3 the polynomial equation given would be 0.... Therefore they are called roots of the polynomial (equation).

Now to understand the part about the sign changing... just plug values in. Start with the first one.

If x > 3. ... say x = 4... x-3 = 1,x-2 = 2, x-1 = 3

1.2.3 = 6 > 0

And so on... so you can see that any value > 3 will give you a positive result.

Take a value between 2 and 3. Say 2.5

x-3 = -0.5, x-2 = 0.5, x-1 = 1.5

You will multiply one negative with 2 positives. Therefore the result will always be negative.

Take a value between 1 and 2. Say 1.5

x-3 = -1.5, x-2= -0.5, x-1 = 0.5

You will multiply 2 negatives and one positive. So the result will always be positive.

Take a number less than 1. Say 0.5

This time you will multiply 3 negative numbers after doing the subtractions... meaning the result will always be negative.

You can see that the sign changes around the roots... namely 1,2,and,3

So if you want a positive result when using the equation above...

The number you choose will have to be between 1 and 2 (not including 1 and 2 as they are roots and will make the value of the equation 0) and of course (3 to +ve infinity)

If you want to represent a set of numbers and not include the boundaries, you put them within parens.

(1,2) means all numbers greater than 1 but less than 2

(3, inf) means all numbers greater than 3 upto infinity.