Roots of Equations


What?
The roots of an equation is what x equals when f(x)=0. You can solve for these algebraically or graphically. Sometimes it is hard to see exact values on a graph so solving for x algebraically is the preferred method.

The Root of Roots (of Equations)
People began to study polynomial equations way back in Babylonian days. They would use polynomials to solve for small powers and specific coefficients. There is also evidence that the Egyptians used the Quadratic Formula in order to solve problems with only a couple exponents. These people were very limited with what they could do with polynomials because they did not yet understand imaginary, complex, nor negative numbers. They would not begin to comprehend the complex numbers until after 500 B.C. This was when the Pythagoreans solved the Pythagoras Theorem and discovered that some equations, like x^2=2, do not have a rational root (Asiru, 2005).

Though time many mathematicians have tried to come up with formulas like the Quadratic Formula for higher exponential equations. A group of them succeeded in the 1500’s for equations with the powers of 3 and 4. There were many failed attempts to find formulas for the higher functions until the 18th century when Ruffini and Able discovered a proof that there is cannot be a formula for any equation with exponents greater than 4 (Pan, 1997). Through the failed attempts many other things surfaced. A couple of examples of this for cubic equations are 1) that there are only two options for their roots. We know that either all the roots are all rational numbers or that there are two irrational numbered roots and only one rational number root (Grewal & Pathak, 2002). And 2) that given a set of predetermined conditions, one root of the cubic function is always a negative reciprocal of another (Asiru, 2005).

Descartes is one of the distinguished mathematicians that made advancements in finding the roots of equations. He bridged the gap between geometry and algebra when studying the roots of equations as he used them to describe the continuous motion of geometrical curves (The history of, 1987). In his book La Geometrie Descartes took advantage of the fact that taking all the terms to the same side of the equation and leave only a zero on the other makes the equation easier to work with. He made note that if a is a zero of the equation than it is divisible by the factor (x-a). Descartes discovered for an equation to the nth degree, there were n roots. His findings influenced the work of many mathematicians of his day and still continue to have influence on how we do things today (Burton, 2010).

Today the solving for equations is done in different fields of science, engineering, business, and statistics but they are being simplified down to linear algebra and programing. Because of this the main area that still uses roots is computer algebra. In this field they sometimes use equations with an order over 100! They are to constantly seeking new algorithms to implement into our computer software that our society is so dependent on (Asiru, 2005).

How to Find Roots
Find the roots to a bimonial equation:
  1. Simplify the equation and write in standard form
  2. Factor and then move on to step 3 or use the Quadratic Formula to find what x equals
  3. If factor, then set each factor equal to 0 and solve for x
  4. Check answers graphically and algebraically

Find the roots to a polynomial equation:
  1. Simplify the equation and write in standard form
  2. Use Rational Roots Test to find all the possibilties for real roots
  3. Use Polynomial Long Division or Synthetic Division to take that factor out of the equation
  4. Repeat step 3 until you have found all the real factors of x, or are down to a binomial equation
  5. Now you can use the steps for finding binomial roots to complete the problem
  6. Check answers graphically and algebraically


Teaching the Roots of Equations Activity
Dividing an Equation by its Roots Applet

Resources