Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  2x3 -  32x  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   2x³ - 32x  =   2x • (x² - 16) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  x² - 16 

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 : 16 is the square of 4
Check :  x²  is the square of  x¹ 

Factorization is :       (x + 4)  •  (x - 4) 

Equation at the end of step  3  :

  2x • (x + 4) • (x - 4)  = 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  :    2x = 0 

 Divide both sides of the equation by 2:
                     x = 0

Solving a Single Variable Equation :

 4.3      Solve  :    x+4 = 0 

 Subtract  4  from both sides of the equation : 
                      x = -4

Solving a Single Variable Equation :

 4.4      Solve  :    x-4 = 0 

 Add  4  to both sides of the equation : 
                      x = 4

Three solutions were found :

1.  x = 4
2.  x = -4
3.  x = 0