Factor table with sign-The useful tool to solve polynomial inequality

First of all, let's try these questions.

Question 1.
Find the integer value of \(x\) that \((x-2)(x-4)<0\)

Question 2.
Solve for this inequation: \(x(x+1)(x-5)>0\)

Question 3.
Solve for this inequation: \(\frac{x(x-2)^3}{(x+1)(x+2)^2} \leq 0\)

To solve Question 1, we could quickly come to result \(x=3\). However, to solve Question 2 and Question 3, how much time will we need? Almost much more time to solve them than to solve Question 1.

Basically, while solving polynomial inequalitie, we need to review each range value of \(x\).

For example, to solve Question 1: \((x-2)(x-4)<0\), we need to review if \(x<2\), if \(2 \leq x<4\), and \(x \geq 4\). Finally, we come to result \(2 <x<4\).

This method could be used with basic polynomial inequalitie, but it seems unuseful to solve complex polynomial inequalitie. Now, I'm going to introduce you an useful tool to solve these kinds of complex polynomial inequalitie, called: Factor table with sign.

These are the steps to solve polynomial inequalities like \((x-a_1)(x-a_2)...(x-a_n) < 0\). (note that sign "\(<\)" could be "\(>\)", "\(\leq\)", "\(\geq\)")

Step 1. Change the polynomial/expression into factorization form.
Step 2. Make factor table with sign to solve the inequatility
Step 2.1. Find all values of \(x\) those make each factor equals to 0. List them in the first row of table in ascending order.
Step 2.2. In next rows of table, review the signs of each factor of the polynomial/expression based on each range value of \(x\)
Step 2.3. In the fianl row of table, review the sign of the polynomial/expression based on each sign of each factor in each range value of \(x\).
Step 3. Finally, come to solution.

Now, let's solve the Question 1.
Step 1. We already have \((x-2)(x-4)<0\)
Step 2. Make factor table with sign like this.


Step 2.1. Find all values of \(x\) those make each factor equals to 0.
\(x-2=0 \iff x=2\)
\(x-4=0 \iff x=4\)
In the first row, list all these values in ascending order: \(-\infty , 2, 4, +\infty\).

Step 2.2. In next 2 rows, review the signs of each factor.
This step based simply on this rule: If \(x<a \implies x-a<0\). If \(x>a \implies x-a>0\).

For example, \(x-2=0 \implies x=2\). Hence, any value of \(x<2\), then \(x-2\) will be negative (-); any value of \(x>2\), then \(x-2\) will be positive (+).

We dont need to care about what is the value of \(x-2\) if \(x=4\) or what is the value of \(x-4\) if \(x=2\). These values are expressed as sign "|".

Step 2.3. In final row, review the sign of the expression.
The sign of the expression simply based on this rule:
\((+) \times (+) = (+) \)
\((-) \times (-) = (+) \)
\((+) \times (-) = (-) \)
\((-) \times (+) = (-) \)

Step 3. Come to solution.
We need to find sign (-) of the expression. Based on the table, we simply have \(2<x<4\).

That's it.

Now let's solve Question 2.
\(x=0\)
\(x+1=0 \implies x=-1\)
\(x-5=0 \implies x=5\)

The factor table with sign:

Hence, we simply have \(x \in (-1,0) \cup (5, +\infty)\)


Solution for Question 3.
\(x=0\)
\(x-2=0 \implies x=2\)
\(x+1=0 \implies x=-1\)
\(x+2=0 \implies x=-2\)

The factor table with sign


In the table above, I've divided into 4 parts. Part 2 contains every factor without power for simplified purpose. Then I come to Part 3 that contains every factor with its power. In part 4 or the final row, I come to the sign of the expression.

Also note that the expression is undefined when \(x=-2\) or \(x=-1\). The solution is \(x \in (-\infty, -2) \cup (-2,-1) \cup [0,2]\)

Now come to another example.
Question 4. Find the product of the integer values of x that satisfy the inequality \(\frac{(x^2-4)^3}{(x-5)^5(x^2-9)^4}\)
This question belongs to e-GMAT https://gmatclub.com/forum/wavy-line-met ... l#p1727256
You could find the solution in https://gmatclub.com/forum/wavy-line-met ... l#p1771328


Question 5. Find the value of \(x\) that \(x^4+5x^3-7x^2-41x-30 \leq 0\).

Solution.
\(x^4+5x^3-7x^2-41x-30 =(x-3)(x+1)(x+2)(x+5)\)

The factor table with sign

The solution is \(x \in [-5,-2] \cup [-1,3]\)


This tool is really useful when we need to find the value of x satisfied the polynomial inequalities. I hope this tool could assist you in solve DS/PS GMAT questions in the future.
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