Solution - Linear equations with one unknown

Step by step solution :

Step  1  :

Checking for a perfect cube :

 1.1    t³-t²+t-1  is not a perfect cube

Trying to factor by pulling out :

 1.2      Factoring:  t³-t²+t-1 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  t-1 
Group 2:  t³-t² 

Pull out from each group separately :

Group 1:   (t-1) • (1)
Group 2:   (t-1) • (t²)
               -------------------
Add up the two groups :
               (t-1)  •  (t²+1) 
Which is the desired factorization

Polynomial Roots Calculator :

 1.3    Find roots (zeroes) of :       F(t) = t²+1
Polynomial Roots Calculator is a set of methods aimed at finding values of  t  for which   F(t)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  t  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  1  :

  (t2 + 1) • (t - 1)  = 0 

Step  2  :

Theory - Roots of a product :

 2.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

 2.2      Solve  :    t²+1 = 0 

 Subtract  1  from both sides of the equation : 
                      t² = -1
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      t  =  ± √ -1  

 In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 
The equation has no real solutions. It has 2 imaginary, or complex solutions.

                      t=  0.0000 + 1.0000 i 
                      t=  0.0000 - 1.0000 i 

Solving a Single Variable Equation :

 2.3      Solve  :    t-1 = 0 

 Add  1  to both sides of the equation : 
                      t = 1

Three solutions were found :

1.  t = 1
2.   t=  0.0000 - 1.0000 i 
   
3.   t=  0.0000 + 1.0000 i