7. Sign Charts for Products

A sign chart tracks whether each factor is positive or negative in each interval.

Example:
\[ (x-1)(x+2)(x-4)\ge0. \]
Critical points:
\[ x=-2,\quad x=1,\quad x=4. \]
They split the number line:

• \((-\infty,-2)\)
• \((-2,1)\)
• \((1,4)\)
• \((4,\infty)\)

Test one value in each interval:

• \(x=-3\): \((-)(-)(-)\) = negative.
• \(x=0\): \((-)(+)(-)\) = positive.
• \(x=2\): \((+)(+)(-)\) = negative.
• \(x=5\): \((+)(+)(+)\) = positive.

Because the inequality is \(\ge0\), include positive intervals and zero points:
\[ [-2,1]\cup[4,\infty). \]

Repeated factors matter. For \((x-2)^2\), the sign does not change when crossing \(x=2\), because a square is never negative.

Common mistake: assuming signs alternate every time. They alternate at simple roots, but not at even powers.