Solution - Power equations

Step  1  :

Equation at the end of step  1  :

  
 Step  2  :

Equation at the end of step  2  :

  
 Step  3  :

Equation at the end of step  3  :

  
 Step  4  :

Equation at the end of step  4  :

Step  5  :

            30x3 - 5x244
 Simplify   ————————————
             5x2 - 5x6  

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   30x³ - 5x²⁴⁴  =   5x³ • (6 - x²⁴¹) 

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   5x² - 5x⁶  =   -5x² • (x⁴ - 1) 

Trying to factor as a Difference of Squares :

 7.2      Factoring:  x⁴ - 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 :  x⁴  is the square of  x² 

Factorization is :       (x² + 1)  •  (x² - 1) 

Polynomial Roots Calculator :

 7.3    Find roots (zeroes) of :       F(x) = x² + 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  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 :

 7.4      Factoring:  x² - 1 

Check : 1 is the square of 1
Check :  x²  is the square of  x¹ 

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

Dividing exponential expressions :

 7.5    x³ divided by x² = x^{(3 - 2)} = x¹ = x  7.6      Attempting Polynomial Long Division

   Attempted Long division of
      6 - x²⁴¹ 
   By :
      -x² - 1 
   Was aborted due to the followinf reason :

   Power of dividend too high for Tiger to process

Step  8  :

Pulling out like terms :

 8.1     Pull out like factors :

   -x² - 1  =   -1 • (x² + 1) 

Polynomial Roots Calculator :

 8.2    Find roots (zeroes) of :       F(x) = x² + 1

     See theory in step 7.3
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

Final result :

          x • (6 - x241)       
  —————————————————————————————
  (-x2 - 1) • (x + 1) • (x - 1)