Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (((x4) -  2x3) -  x2) +  2x  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   x⁴ - 2x³ - x² + 2x  = 

  x • (x³ - 2x² - x + 2) 

Checking for a perfect cube :

 3.2    x³ - 2x² - x + 2  is not a perfect cube

Trying to factor by pulling out :

 3.3      Factoring:  x³ - 2x² - x + 2 

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

Group 1:  -x + 2 
Group 2:  x³ - 2x² 

Pull out from each group separately :

Group 1:   (-x + 2) • (1) = (x - 2) • (-1)
Group 2:   (x - 2) • (x²)
               -------------------
Add up the two groups :
               (x - 2)  •  (x² - 1) 
Which is the desired factorization

Trying to factor as a Difference of Squares :

 3.4      Factoring:  x² - 1 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 1 is the square of 1
Check :  x²  is the square of  x¹ 

Factorization is :       (x + 1)  •  (x - 1) 

Equation at the end of step  3  :

  x • (x + 1) • (x - 1) • (x - 2)  = 0 

Step  4  :

Theory - Roots of a product :

 4.1    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 :

 4.2      Solve  :    x = 0 

  Solution is  x = 0

Solving a Single Variable Equation :

 4.3      Solve  :    x+1 = 0 

 Subtract  1  from both sides of the equation : 
                      x = -1

Solving a Single Variable Equation :

 4.4      Solve  :    x-1 = 0 

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

Solving a Single Variable Equation :

 4.5      Solve  :    x-2 = 0 

 Add  2  to both sides of the equation : 
                      x = 2

Four solutions were found :

1.  x = 2
2.  x = 1
3.  x = -1
4.  x = 0