Solution - Adding, subtracting and finding the least common multiple

Step  1  :

            1
 Simplify   —
            x

Equation at the end of step  1  :

                 8      1
  (((((x2)+2x)-————)-x)+—)+4
               (x2)     x
 Step  2  :
             8
 Simplify   ——
            x2

Equation at the end of step  2  :

                8     1
  (((((x2)+2x)-——)-x)+—)+4
               x2     x

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  x²  as the denominator :

                x2 + 2x     (x2 + 2x) • x2
     x2 + 2x =  ———————  =  ——————————————
                   1              x2      

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   x² + 2x  =   x • (x + 2) 

Adding fractions that have a common denominator :

 4.2       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:

 x • (x+2) • x2 - (8)     x4 + 2x3 - 8
 ————————————————————  =  ————————————
          x2                   x2     

Equation at the end of step  4  :

    (x4 + 2x3 - 8)          1     
  ((—————————————— -  x) +  —) +  4
          x2                x     

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

         x     x • x2
    x =  —  =  ——————
         1       x2  

Polynomial Roots Calculator :

 5.2    Find roots (zeroes) of :       F(x) = x⁴ + 2x³ - 8
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  -8.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions

 (x4+2x3-8) - (x • x2)     x4 + x3 - 8
 —————————————————————  =  ———————————
          x2                   x2     

Equation at the end of step  5  :

   (x4 + x3 - 8)    1     
  (————————————— +  —) +  4
        x2          x     

Step  6  :

Polynomial Roots Calculator :

 6.1    Find roots (zeroes) of :       F(x) = x⁴+x³-8

     See theory in step 5.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -8.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8

 Let us test ....

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that
   x⁴+x³-8 
can be divided with  x+2 

Polynomial Long Division :

 6.2    Polynomial Long Division
Dividing :  x⁴+x³-8 
                              ("Dividend")
By         :    x+2    ("Divisor")

Quotient :  x³-x²+2x-4  Remainder:  0 

Polynomial Roots Calculator :

 6.3    Find roots (zeroes) of :       F(x) = x³-x²+2x-4

     See theory in step 5.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -4.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4

 Let us test ....

Polynomial Roots Calculator found no rational roots

Calculating the Least Common Multiple :

 6.4    Find the Least Common Multiple

      The left denominator is :       x² 

      The right denominator is :       x 

      Least Common Multiple:
      x² 

Calculating Multipliers :

 6.5    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 = 1

   Right_M = L.C.M / R_Deno = x

Making Equivalent Fractions :

 6.6      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.      (x3-x2+2x-4) • (x+2)
   ——————————————————  =   ————————————————————
         L.C.M                      x2         

   R. Mult. • R. Num.       x
   ——————————————————  =   ——
         L.C.M             x2

Adding fractions that have a common denominator :

 6.7       Adding up the two equivalent fractions

 (x3-x2+2x-4) • (x+2) + x     x4 + x3 + x - 8
 ————————————————————————  =  ———————————————
            x2                      x2       

Equation at the end of step  6  :

  (x4 + x3 + x - 8)    
  ————————————————— +  4
         x2            

Step  7  :

Rewriting the whole as an Equivalent Fraction :

 7.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

         4     4 • x2
    4 =  —  =  ——————
         1       x2  

Checking for a perfect cube :

 7.2    x⁴ + x³ + x - 8  is not a perfect cube

Trying to factor by pulling out :

 7.3      Factoring:  x⁴ + x³ + x - 8 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  x - 8 
Group 2:  x⁴ + x³ 

Pull out from each group separately :

Group 1:   (x - 8) • (1)
Group 2:   (x + 1) • (x³)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

 7.4    Find roots (zeroes) of :       F(x) = x⁴ + x³ + x - 8

     See theory in step 5.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -8.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 7.5       Adding up the two equivalent fractions

 (x4+x3+x-8) + 4 • x2     x4 + x3 + 4x2 + x - 8
 ————————————————————  =  —————————————————————
          x2                       x2          

Polynomial Roots Calculator :

 7.6    Find roots (zeroes) of :       F(x) = x⁴ + x³ + 4x² + x - 8

     See theory in step 5.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -8.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  x4 + x3 + 4x2 + x - 8
  —————————————————————
           x2