# What is a sign chart?

Jan 16, 2017

#### Explanation:

Sign chart is used to solve inequalities relating to polynomials, which can be factorized into linear binomials. For example, of the type

$\left(a x + b\right) \left(g x + h\right) \left(p x + q\right) \left(s x + t\right) > 0$

It could also be less than or less than or equal or greater than or equal, but the process is not much effected.

Note that these can be written as

$\left(x - \alpha\right) \left(x - \beta\right) \left(x - \gamma\right) \left(x - \delta\right) > 0$

(here $\alpha = - \frac{b}{a}$, $\beta = - \frac{h}{g}$, $\gamma = - \frac{q}{p}$ and $\delta = - \frac{t}{s}$)

Note that numbers $\alpha$, $\beta$, $\gamma$ and $\delta$ divide real number in five intervals. (Note at these values, value of polynomial will be zero.)

For example, if they are already in increasing order, these are $\left(- \infty , \alpha\right)$, $\left(\alpha , \beta\right)$, beta,gamma), $\left(\gamma , \delta\right)$ and $\left(\delta . \infty\right)$.

In these intervals, we can find that each of these linear binomial i.e. $\left(x - \alpha\right)$, $\left(x - \beta\right)$, $\left(x - \gamma\right)$ and $\left(x - \delta\right)$ take up either a positive or negative value,

and hence the polynomial (as it is a product of these linear binomials) will take positive or negative value

and can easily check the intervals, where the inequality is satisfied, giving us the result.

As an example, one may like to see solution to this question. .