Solving Polynomial Inequality by Mapping

Today let’s tackle polynomial inequality by curve mapping. We don’t need to test the intervals to check if it satisfies the polynomial inequalities. This could be the fastest way for me to solve this type of problem. Before we start to discuss the method it is important for us to know some terms to be used here.

Important Points

Turning points – In a given polynomial equation with degree n the number of turning points is n-1

Trend – like linear functions, polynomial functions has trend as well.

Given a polynomial function f(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+ax+c

If n is even and a_n is positive, both ends of the curve is facing upward.polynomial1

If n is even and a_n is negative, both ends of the curve is facing downward.polynomial2

If n is odd and a_n is positive the trend of the graph is increasing. If the graph is increasing, the graph can be traced from quadrant 3 up to quadrant 1.polynomial3

If n is odd and a_n is negative the trend of the graph is decreasing.If the graph is decreasing, the graph can be traced from quadrant 4 up to quadrant 2.polynomial4

To illustrate the mapping method, let’s provide an example.

Problem:

Solve the solution set of inequality (x-1)(x+2)(x-3)>0

1. Solve the real roots of equation.

The roots are: \{ 1,-2,3\}

2. Draft the roots in coordinate system. The points are (1,0),(-2,0),(3,0)tracing1

3. Take note of the four important points above upon tracing the curve.tracing2

4. The inequality symbol “>” means that the line is above x-axis. The red lines on the curve are the intervals in which the curve is greater than 0 or above x-axis.tracing3

 Worked Problem 1:

Find the solution set of the inequality x^3-4x^2-5x-2\le 0

Solution:

Solve the roots of equation as is, don’t mind the inequality symbol this time. Using synthetic division we have x={2,1(double root)}

Repeat the process mentioned above.

Here is the graph of equation. The solution set is (-\infty,2]question1poly

 

Worked Problem 2:

For what values of x such that the graph of the function f(x)=x^3-4x^2-x+2 below x-axis.

Solution:

The roots of equation are \{-1,1,2\}. F(x)<0  if  (-\infty,-1)\cup(1,2)

Worked Problem 3:

For what values of x does the graph of quartic f(x)=(x-1)^2(x-2)^2 above x-axis?

Solution:

Above x-axis means that y>0 making the equation (x-1)^2(x-2)^2 >0. Since the leading coefficient is greater than 0. Two ends of the graph should be facing upward. The graph also tangent to x-axis through points (1,0) and (2,0) with 3 turning points. Draw a quick draft of the curve and check the values of x in which the y>0. By inspection, y>0 in all real values of x except 1 and 2.

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