Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x7"   was replaced by   "x^7". 

Rearrange:

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

                     (x^4)/3-x/7-((x^7)/5)=0 

Step by step solution :

Step  1  :

            x7
 Simplify   ——
            5 

Equation at the end of step  1  :

   (x4)    x     x7
  (———— -  —) -  ——  = 0 
    3      7     5 

Step  2  :

            x
 Simplify   —
            7

Equation at the end of step  2  :

   (x4)    x     x7
  (———— -  —) -  ——  = 0 
    3      7     5 
 Step  3  :
            x4
 Simplify   ——
            3 

Equation at the end of step  3  :

   x4    x     x7
  (—— -  —) -  ——  = 0 
   3     7     5 

Step  4  :

Calculating the Least Common Multiple :

 4.1    Find the Least Common Multiple

      The left denominator is :       3 

      The right denominator is :       7 

      Least Common Multiple:
      21 

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 = 7

   Right_M = L.C.M / R_Deno = 3

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.      x4 • 7
   ——————————————————  =   ——————
         L.C.M               21  

   R. Mult. • R. Num.      x • 3
   ——————————————————  =   —————
         L.C.M              21  

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:

 x4 • 7 - (x • 3)     7x4 - 3x
 ————————————————  =  ————————
        21               21   

Equation at the end of step  4  :

  (7x4 - 3x)    x7
  —————————— -  ——  = 0 
      21        5 

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   7x⁴ - 3x  =   x • (7x³ - 3) 

Trying to factor as a Difference of Cubes:

 6.2      Factoring:  7x³ - 3 

Theory : A difference of two perfect cubes,  a³ - b³ can be factored into
              (a-b) • (a² +ab +b²)

Proof :  (a-b)•(a²+ab+b²) =
            a³+a²b+ab²-ba²-b²a-b³ =
            a³+(a²b-ba²)+(ab²-b²a)-b³ =
            a³+0+0-b³ =
            a³-b³

Check :  7  is not a cube !!

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

Polynomial Roots Calculator :

 6.3    Find roots (zeroes) of :       F(x) = 7x³ - 3
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  7  and the Trailing Constant is  -3.

 The factor(s) are:

of the Leading Coefficient :  1,7
 of the Trailing Constant :  1 ,3

 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 :       21 

      The right denominator is :       5 

      Least Common Multiple:
      105 

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 = 5

   Right_M = L.C.M / R_Deno = 21

Making Equivalent Fractions :

 6.6      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      x • (7x3-3) • 5
   ——————————————————  =   ———————————————
         L.C.M                   105      

   R. Mult. • R. Num.      x7 • 21
   ——————————————————  =   ———————
         L.C.M               105  

Adding fractions that have a common denominator :

 6.7       Adding up the two equivalent fractions

 x • (7x3-3) • 5 - (x7 • 21)     -21x7 + 35x4 - 15x
 ———————————————————————————  =  ——————————————————
             105                        105        

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   -21x⁷ + 35x⁴ - 15x  =   -x • (21x⁶ - 35x³ + 15) 

Trying to factor by splitting the middle term

 7.2     Factoring  21x⁶ - 35x³ + 15 

The first term is,  21x⁶  its coefficient is  21 .
The middle term is,  -35x³  its coefficient is  -35 .
The last term, "the constant", is  +15 

Step-1 : Multiply the coefficient of the first term by the constant   21 • 15 = 315 

Step-2 : Find two factors of  315  whose sum equals the coefficient of the middle term, which is   -35 .

For tidiness, printing of 18 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  7  :

  -x • (21x6 - 35x3 + 15)
  ———————————————————————  = 0 
            105          

Step  8  :

When a fraction equals zero :

 8.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  -x•(21x6-35x3+15)
  ————————————————— • 105 = 0 • 105
         105       

Now, on the left hand side, the  105  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -x  •  (21x⁶-35x³+15)  = 0

Theory - Roots of a product :

 8.2    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 :

 8.3      Solve  :    -x = 0 

 Multiply both sides of the equation by (-1) :  x = 0

Solving a Single Variable Equation :

Equations which are reducible to quadratic :

 8.4     Solve   21x⁶-35x³+15 = 0

This equation is reducible to quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using  w , such that  w = x³  transforms the equation into :
 21w²-35w+15 = 0

Solving this new equation using the quadratic formula we get two imaginary solutions :
   w = 367.5000 ± 62.1188 i 
Now that we know the value(s) of  w , we can calculate  x  since  x  is  ∛ w  

Since we are speaking 3rd root, each of the two imaginary solutions of has 3 roots

Tiger finds these roots using de Moivre's Formula

The 3rd roots of  367.500 + 62.119 i   are:

  x =  7.185 + 0.401 i 
  x = -3.940 + 6.022 i 
  x = -3.245 -6.423 i 

3rd roots of  367.500-62.119 i  :
  x = -3.245  + 6.423 i  
  x = -3.940 - 6.022 i 
  x =  7.185 - 0.401 i 
 

7 solutions were found :

1.   x = 7.185 - 0.401 i
2.   x = -3.940 - 6.022 i
3.   x = -3.245 + 6.423 i
4.   x = -3.245 -6.423 i
5.   x = -3.940 + 6.022 i
6.   x = 7.185 + 0.401 i
7.  x = 0