Chapter 3

1st Degree Polynomial Equations
ax+b=0

3.4. First Degree Polynomial Equations with Parameters

In the previous examples of this chapter, every time we ended up with an equation of the form ax+b=0 where a and b were specific numbers. We could instantly then see in which case of the table in paragraph 3.1 our equation corresponded.
This is not valid though if the coefficients a,b are presented with the help of letters. In this case these letters are called parameters. Solving equations with parameters is a little more harder.

Example 1

Solve for the various values of  the equation  .

Solution

The equation can be solved as follows:

(1)

We have the following possibilities:

A) If  and  then

B) If then (1)0x= -1 which does not have a solution in R

C) If  then (1)0x=0 which is true for every

Example 2

Prove that for the every value of the parameters a,b the equation bellow has solution:

Solution

(1)

Now we have the following possibilities:

A) If then (1) has unique solution and that is:
B) If a+b=0 then (1) becomes 0x=0 which has infinitive number of solutions

That means that in every case the equation has a solution.

Exercise 1

For the various values of the parameters  solve the equation: