Question 1141444
<br>
In the solution by tutor @mathlover1, she identified the values of x that make each of the individual factors negative or zero; but it is unclear how she used that information to identify the intervals on which the complete expression is negative or zero.<br>
Consider the polynomial function<br>
{{{f(x) = (x+5)(x-2)(x-7)}}}<br>
We want to identify when the function value is zero or negative.<br>
Clearly the function value is zero at x=-5, x=2, and x=7.<br>
To identify the intervals on which the function value is negative, there are several options.  Among those options are<br>
(1) The function is a cubic with positive leading coefficient.  Think of what the graph of that kind of function looks like.  The function value is negative for "large negative" value of x and positive for large positive values of x.  That, along with the known zeros of the function, will determine the intervals on which the function value is negative.<br>
(2) Choose a test point in each of the intervals determined by the zeros of the function to see whether the function value is positive or negative in each interval.<br>
(3) (My personal choice....) You know that for large negative values of x all three factors are negative, so the function value is negative to the left of x=-5.  Then, as you "walk" along the number line to the right, the function value changes sign each time you pass one of the zeros.  So the function value is (a) negative for x less than -5; (b) positive for x between -5 and 2; (c) negative for x between 2 and 7; and (d) positive for x greater than 7.