Question

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Answer 1

The first term is,  x2  its coefficient is  1 .
The middle term is,  -5x  its coefficient is  -5 .
The last term, "the constant", is  -104 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -104 = -104 

Step-2 : Find two factors of  -104  whose sum equals the coefficient of the middle term, which is   -5 .

     -104   +   1   =   -103     -52   +   2   =   -50     -26   +   4   =   -22     -13   +   8   =   -5   That's it


Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -13  and  8 
                     x2 - 13x + 8x - 104

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-13)
              Add up the last 2 terms, pulling out common factors :
                    8 • (x-13)
Step-5 : Add up the four terms of step 4 :
                    (x+8)  •  (x-13)
             Which is the desired factorization

Equation at the end of step  1  :

(x + 8) • (x - 13) = 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  :    x+8 = 0 

 Subtract  8  from both sides of the equation : 
                      x = -8 

Solving a Single Variable Equation :

 2.3      Solve  :    x-13 = 0 

 Add  13  to both sides of the equation : 
                      x = 13