Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Trying to factor as a Difference of Squares :

 1.1      Factoring:  z⁸-1 

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 :  z⁸  is the square of  z⁴ 

Factorization is :       (z⁴ + 1)  •  (z⁴ - 1) 

Polynomial Roots Calculator :

 1.2    Find roots (zeroes) of :       F(z) = z⁴ + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  z  for which   F(z)=0  

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

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares :

 1.3      Factoring:  z⁴ - 1 

Check : 1 is the square of 1
Check :  z⁴  is the square of  z² 

Factorization is :       (z² + 1)  •  (z² - 1) 

Polynomial Roots Calculator :

 1.4    Find roots (zeroes) of :       F(z) = z² + 1

     See theory in step 1.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  1.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares :

 1.5      Factoring:  z² - 1 

Check : 1 is the square of 1
Check :  z²  is the square of  z¹ 

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

Equation at the end of step  1  :

  (z4 + 1) • (z2 + 1) • (z + 1) • (z - 1)  = 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  :    z⁴+1 = 0 

 Subtract  1  from both sides of the equation : 
                      z⁴ = -1
                     z  =  ∜ -1  

 The equation has no real solutions. It has 4 imaginary, or complex solutions.

                      z=  0.7071 + 0.7071 i 
                      z=  -0.7071 + 0.7071 i 
                      z=  -0.7071 - 0.7071 i 
                      z=  0.7071 - 0.7071 i 

Solving a Single Variable Equation :

 2.3      Solve  :    z²+1 = 0 

 Subtract  1  from both sides of the equation : 
                      z² = -1
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      z  =  ± √ -1  

 In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 
The equation has no real solutions. It has 2 imaginary, or complex solutions.

                      z=  0.0000 + 1.0000 i 
                      z=  0.0000 - 1.0000 i 

Solving a Single Variable Equation :

 2.4      Solve  :    z+1 = 0 

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

Solving a Single Variable Equation :

 2.5      Solve  :    z-1 = 0 

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

8 solutions were found :

1.  z = 1
2.  z = -1
3.   z=  0.0000 - 1.0000 i 
   
4.   z=  0.0000 + 1.0000 i 
   
5.   z=  0.7071 - 0.7071 i 
   
6.   z=  -0.7071 - 0.7071 i 
   
7.   z=  -0.7071 + 0.7071 i 
   
8.   z=  0.7071 + 0.7071 i