Unit roots/Factorization and Solving Equations

In this chapter, we will find some use of the roots of unity in factorization and solving equations.

Factorization and solving equations

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To solve an equation is to find the set of values of the unknowns satisfying the equation. While we can easily solve equations of lower degrees, it is not easy to solve equations of higher degrees. However, by factorization, we can rewrite a high degree polynomial into a product of low degree polynomials. The method of solving equations be factorization is based on the following theorem:

Zero product property Let   and   be real or complex valued. If  , then   or  .
Proof Either   or   holds true. If  , then "  or  " holds true. If  , then   exists. Multiply it to both sides of  :

 
 .

Then "  or  " also holds true.QED

Therefore, when we solve a polynomial equation, we can first factorize the polynomial and form smaller equations equating the factors with zero.

Example 1 Solve  .
Solution

 
 
 
 
  or   or  
  or   or  
  or  

In the previous chapter, we have seen the use of unit roots in determining the divisibility of a polynomial. In fact we can similarly factorize some polynomials by considering the properties of unit roots.

Example 2 Factorize  .
Anaylsis The indices in the expression are 8, 6, 4, 2, 0. When they are divided by 5, the remainders are 3, 1, 4, 2, 0. Therefore, when   is replaced by a non-real fifth root of unity, the expression equals zero. So the expression is divisible by  .
Answer  

If we allow complex coefficients in the factors, any polynomials can be factorized as a product of linear factors; if we allow any real coefficients, then any polynomials with real coefficients can be factorized as a product of factors of at most the second degree.

Example 3 Factorize
(a)   (three factors)
(b)   (two factors)
Analysis We can check that both (a) and (b) is zero when   is replaced by a non-real cube root of unity. However, we have to further find another factor for (a). For this purpose, we first let   so the expression becomes  , which also equals zero when   is replaced by a non-real cube root of unity.
Solution (a)  . So,

 

(b)  

 
 

The cubic equation

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Using the cube root of unity, we can derive the formula for cubic equations.

Let   be a non-real cube root of unity, then:

 ,  .

Then we can show that:

 .

Therefore, the roots of the cubic equation in  :

 

are:

 ,  ,  .

Now consider the cubic equation:

 .

We may let   and  . So  . Therefore,   and   are the roots of the following equation:

 .

The roots of this equation are:

 .

We let   be any of these roots, and   be the other root. Then:

 ,
 .

(In fact we may also take other non-real cube roots, as long as the relation   holds. However, we need not consider the non-real cube roots because we have considered them through the specification of   and  .)

Example 4 Solve  .
Solution In the example,  ,  . So:

 ,
 .

Therefore, the roots are:

 ,  ,  .

We can verify that  .

Higher degree equations

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We can also use roots of unity to solve some higher degree equations in some special forms.

Example 5 Solve  
Solution From the previous example,  , where:

 
 

From  , we can obtain four roots:

  (i.e. the non-real fifth roots of unity.)

Note that  , Therefore,  . So, the other four roots are:

 .

The original equation has a total of eight roots:

 .

Alternative solution Let  , the equation becomes:

 .

The four roots of this equation are:

 

To obtain the corresponding values of  , we take the square roots of each of these values of  . So, the roots are

 .

Although the form of the roots given by the two methods are different, they are identical set of numbers. Both can be written as:

 .

They are the eight non-real tenth roots of unity.

Example 6 Find a equations with rational coefficients such that its roots equal  , where   is a root of the equation  .