College Algebra & Trigonometry, 2018a

92 CHAPTER 2. POLYNOMIAL AND RATIONAL FUNCTIONS
2.2 Solution by Graphing
In previous courses the solution of linear equations is covered, generally by separating
the variables and constants on opposite sides of the equation to isolate the
variable. In the previous chapter we examined the solution of quadratic equations,
in which the variable is isolated using the technique of completing the
square. The major difference between these methods of solution is that, in the
solving of quadratic equations, we must contend with several different powers
of the variable which makes it considerably more difficult to isolate the variable.
There are formulas like the quadratic formula available to solve cubic (x 3 ) and
quartic (x 4 ) equations, however these formulas are somewhat cumbersome and
archaic. The primary method for solving eqautions of degree higher than 2 is
solution by graphing or by algorithm.
Solution by algorithm is a very interesting process as there are many different
algorithms available. Which algorithm is most appropriate often depends on the
types of equations being solved and the technology available to solve them. Two
major types of algorithms that rely on the graphincal representation of an equation
are called “Double False Position” and “The Newton-Raphson Method.”
Many commonly available pieces of techonolgy use one of these methods. Since
we have graphing calculators available to us, we will focus on solution by graphing.
Whether using a TI (Texas Instruments) or Casio graphing calculator, or a software
based graphing ustility such as Graph or Desmos, the solution of these
equations focuses on finding the x-intercepts of the graph since this is where
the y value is 0. The specific processes for solving equations using each of these
different tools will be covered in class.
Example
Solve for x.
x 3 − 3x 2 =2x − 7
The first step is to move all terms to one side of the equation and set them equal
to zero.
x 3 − 3x 2 − 2x +7=0