Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (((x3) +  3x2) -  16x) -  48  = 0 

Step  2  :

Checking for a perfect cube :

 2.1    x³+3x²-16x-48  is not a perfect cube

Trying to factor by pulling out :

 2.2      Factoring:  x³+3x²-16x-48 

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

Group 1:  x³+3x² 
Group 2:  -16x-48 

Pull out from each group separately :

Group 1:   (x+3) • (x²)
Group 2:   (x+3) • (-16)
               -------------------
Add up the two groups :
               (x+3)  •  (x²-16) 
Which is the desired factorization

Trying to factor as a Difference of Squares :

 2.3      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  2  :

  (x + 4) • (x - 4) • (x + 3)  = 0 

Step  3  :

Theory - Roots of a product :

 3.1    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 :

 3.2      Solve  :    x+4 = 0 

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

Solving a Single Variable Equation :

 3.3      Solve  :    x-4 = 0 

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

Solving a Single Variable Equation :

 3.4      Solve  :    x+3 = 0 

 Subtract  3  from both sides of the equation : 
                      x = -3

Three solutions were found :

1.  x = -3
2.  x = 4
3.  x = -4