Solution - Adding, subtracting and finding the least common multiple

Step  1  :

            1
 Simplify   —
            8

Equation at the end of step  1  :

   3          1
  (— • a) -  (— • a5)
   4          8

Step  2  :

Equation at the end of step  2  :

   3         a5
  (— • a) -  ——
   4         8 

Step  3  :

            3
 Simplify   —
            4

Equation at the end of step  3  :

   3         a5
  (— • a) -  ——
   4         8 

Step  4  :

Calculating the Least Common Multiple :

 4.1    Find the Least Common Multiple

      The left denominator is :       4 

      The right denominator is :       8 

      Least Common Multiple:
      8 

Calculating Multipliers :

 4.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 2

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 4.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)²   and  (y²+y)/(y+1)³  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.      3a • 2
   ——————————————————  =   ——————
         L.C.M               8   

   R. Mult. • R. Num.      a5
   ——————————————————  =   ——
         L.C.M             8 

Adding fractions that have a common denominator :

 4.4       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 3a • 2 - (a5)     6a - a5
 —————————————  =  ———————
       8              8   

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   6a - a⁵  =   -a • (a⁴ - 6) 

Trying to factor as a Difference of Squares :

 5.2      Factoring:  a⁴ - 6 

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 : 6 is not a square !!

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

Polynomial Roots Calculator :

 5.3    Find roots (zeroes) of :       F(a) = a⁴ - 6
Polynomial Roots Calculator is a set of methods aimed at finding values of  a  for which   F(a)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  a  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  -6.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,6

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  +a • (a4 + 6)
  —————————————
        8