Solution - Factoring binomials using the difference of squares

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x8"   was replaced by   "x^8". 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (x2) -  22x8  = 0 
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   x² - 4x⁸  =   -x² • (4x⁶ - 1) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  4x⁶ - 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 :  4  is the square of  2 
Check : 1 is the square of 1
Check :  x⁶  is the square of  x³ 

Factorization is :       (2x³ + 1)  •  (2x³ - 1) 

Trying to factor as a Sum of Cubes :

 3.3      Factoring:  2x³ + 1 

Theory : A sum of two perfect cubes,  a³ + b³ can be factored into  :
             (a+b) • (a²-ab+b²)
Proof  : (a+b) • (a²-ab+b²) =
    a³-a²b+ab²+ba²-b²a+b³ =
    a³+(a²b-ba²)+(ab²-b²a)+b³=
    a³+0+0+b³=
    a³+b³

Check :  2  is not a cube !!

Ruling : Binomial can not be factored as the difference of two perfect cubes

Polynomial Roots Calculator :

 3.4    Find roots (zeroes) of :       F(x) = 2x³ + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  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  2  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1,2
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Cubes:

 3.5      Factoring:  2x³ - 1 

Theory : A difference of two perfect cubes,  a³ - b³ can be factored into
              (a-b) • (a² +ab +b²)

Proof :  (a-b)•(a²+ab+b²) =
            a³+a²b+ab²-ba²-b²a-b³ =
            a³+(a²b-ba²)+(ab²-b²a)-b³ =
            a³+0+0-b³ =
            a³-b³

Check :  2  is not a cube !!

Ruling : Binomial can not be factored as the difference of two perfect cubes

Polynomial Roots Calculator :

 3.6    Find roots (zeroes) of :       F(x) = 2x³ - 1

     See theory in step 3.4
In this case, the Leading Coefficient is  2  and the Trailing Constant is  -1.

 The factor(s) are:

of the Leading Coefficient :  1,2
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  3  :

  -x2 • (2x3 + 1) • (2x3 - 1)  = 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  :    -x² = 0 

 Multiply both sides of the equation by (-1) :  x² = 0


 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 0  

 Any root of zero is zero. This equation has one solution which is  x = 0

Solving a Single Variable Equation :

 4.3      Solve  :    2x³+1 = 0 

 Subtract  1  from both sides of the equation : 
                      2x³ = -1
Divide both sides of the equation by 2:
                     x³ = -1/2 = -0.500
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:  
                      x  =  ∛ -1/2  

 Negative numbers have real cube roots.
 ∛ -1/2 = ∛ -1• 1/2  = ∛ -1 • ∛ 1/2  =(-1)•∛ 1/2 

The equation has one real solution, a negative number This solution is  x = ∛ -0.500 = -0.79370

Solving a Single Variable Equation :

 4.4      Solve  :    2x³-1 = 0 

 Add  1  to both sides of the equation : 
                      2x³ = 1
Divide both sides of the equation by 2:
                     x³ = 1/2 = 0.500
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:  
                      x  =  ∛ 1/2  

 The equation has one real solution
This solution is  x = ∛ 0.500 = 0.79370

Three solutions were found :

1.  x = ∛ 0.500 = 0.79370
2.  x = ∛ -0.500 = -0.79370
3.  x = 0