Solution - Factoring binomials using the difference of squares

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     11-(z*z*z*z)=0 

Step by step solution :

Trying to factor as a Difference of Squares :

 0.1      Factoring:  11 - z⁴ 

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 :  11  is not a square !!

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

Polynomial Roots Calculator :

 0.2    Find roots (zeroes) of :       F(z) = -z⁴ + 11
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  11  and the Trailing Constant is  -1.

 The factor(s) are:

of the Leading Coefficient :  1,11
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Step  1  :

Solving a Single Variable Equation :

 1.1      Solve  :    -z⁴+11 = 0 

 Subtract  11  from both sides of the equation : 
                      -z⁴ = -11
Multiply both sides of the equation by (-1) :  z⁴ = 11


                     z  =  ∜ 11  

 The equation has two real solutions  
 These solutions are  z = ± ∜11 = ± 1.8212  
 

Two solutions were found :