Question

What is a sign chart ?

Answer 1

Hint:Sign charts are a table that indicate the sign of a polynomial between a specific range of values. In that way a sign chart is used to solve inequalities relating to polynomials, which can be factorized into linear binomials. For example, a polynomial of the type \[(ax + b)(gx + h)(px + q)(sx + t) > 0\]. The sign here before zero could also be less than, or less than or equal, or greater than or equal, but the process is not much affected and gives the correct results.

Complete step by step answer:
The above inequality is \[(ax + b)(gx + h)(px + q)(sx + t) > 0\].
It can also be written as,
\[ \Rightarrow (x - \alpha )(x - \beta )(x - \gamma )(x - \delta ) > 0\]
Where \[\;\alpha = - \dfrac{b}{a}\] , \[\beta = - \dfrac{h}{g}\] , \[\gamma = - \dfrac{q}{p}\] and \[\delta = - \dfrac{t}{s}\].

Now, the value of the above polynomial will be zero at \[\alpha ,\beta ,\gamma ,\delta \] .
Therefore the values \[\alpha ,\beta ,\gamma ,\delta \] divide the real numbers in five intervals. For example, if they are in increasing order, then the five obtained intervals are \[\;\left( { - \infty ,\alpha } \right),\;\left( {\alpha ,\beta } \right),(\beta ,\gamma ),\;\left( {\gamma ,\delta } \right),\;\left( {\delta .\infty } \right)\] .

In these five intervals, we can find the signs of each linear binomial among \[(x - \alpha ),(x - \beta ),(x - \gamma ),(x - \delta )\] whether it is positive or negative.And hence, the polynomial \[(x - \alpha )(x - \beta )(x - \gamma )(x - \delta )\] , as it is a product of these linear binomials, will take either positive or negative value between those intervals.And then we can easily check the intervals, where the inequality is satisfied, giving us the required solutions.

Now consider \[\alpha < \beta < \gamma < \delta \]. The sign chart of the above polynomial between those five intervals can be shown as:

Interval \[ \to \]Polynomial \[ \downarrow \]	\[\;\left( { - \infty ,\alpha } \right)\]	\[\left( {\alpha ,\beta } \right)\]	\[(\beta ,\gamma )\]	\[\left( {\gamma ,\delta } \right)\;\]	\[\left( {\delta .\infty } \right)\]
\[(x - \alpha )\]	-ve	+ve	+ve	+ve	+ve
\[(x - \beta )\]	-ve	-ve	+ve	+ve	+ve
\[(x - \gamma )\]	-ve	-ve	-ve	+ve	+ve
\[(x - \delta )\]	-ve	-ve	-ve	-ve	+ve
\[(x - \alpha )(x - \beta )(x - \gamma )(x - \delta )\]	+ve	-ve	+ve	-ve	+ve

This is the required sign chart for a polynomial of type \[(x - \alpha )(x - \beta )(x - \gamma )(x - \delta )\].

Note:Further, now we can also find the solutions of a given inequality by using the above sign chart. Let the inequality be given as
\[ \Rightarrow (x - \alpha )(x - \beta )(x - \gamma )(x - \delta ) > 0\]
Then the polynomial is >0 i.e. positive only in the three intervals \[\;\left( { - \infty ,\alpha } \right)\] , \[(\beta ,\gamma )\] , \[\left( {\delta .\infty } \right)\]. Therefore the solutions of the inequality \[(x - \alpha )(x - \beta )(x - \gamma )(x - \delta ) > 0\] is given by,
\[x < \alpha \] , \[\beta < x < \lambda \] and \[x > \delta \] .