Integral Zero Theorem

The Integral Zero theorem has wide applications in finding the possible roots of a polynomial equation. Apart from the Integral Zero Theorem; Factor Theorem and Rational Zero Theorem are few other theorems that are used to find the possible roots of an equation. Once we have discussed the Integral Zero Theorem, we will take a brief glimpse into these two theorems as well.

 

Theorem of Integral Zero:

 

The Integral Zero Theorem states that:

The integer A is the root of the equation F(x) = ax+ bx + c IF and ONLY IF, A is a factor of the constant term c.  (Constant term being a term in the equation of x without x in it)

For example, consider the equation

F(x) = x2 + 18x + 9

Here the constant term is 9 and a possible root is -3 (being a factor of 9).

 

A Corollary to Integral Zero Theorem:

 

Factor Theorem and Rational Zero Theorem: a corollary to integral zero theorem

Factor Theorem states that the polynomial (x-A) is a divisor (and factor) of the polynomial F(x) IF and ONLY IF A is a root or zero of F(x) 

 Examples of the above.

 ex 1. (x2 + 6x + 9)/(x + 3) = (x + 2) + 0,

 because x = -3 is a root of y=x2 + 6x + 9

 Rational Zero Theorem:The rational fraction A/B is a root of the polynomial F(x) IF and ONLY IF A is a factor of the constant term and B is a factor of the leading coefficient.

 

What is the Root of an Equation?

 

The root of an equation is that value of the variable (usually x), which when substituted in the equation would give 0.

For example, consider the equation

x2 - 9x + 18 = 0

This equation has two roots: 3 and 6, Substituting both in the equation gives 0.

Let us verify, 

put x = 3, the equation can be rewritten as  3- 9(3) + 18,

this becomes 9 - 27 + 3, which is 0

similarly x = 6, the equation will be 6- 9(6) + 18, which again is 0.