Solution - Reducing fractions to their lowest terms

Step  1  :

            28
 Simplify   ——
            x2

Equation at the end of step  1  :

         7          28
  (((x+————)+3x)-10)——)-25)
       (x2)((((x^2)+x2

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

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

                x2 + 3x     (x2 + 3x) • x2
     x2 + 3x =  ———————  =  ——————————————
                   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  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   x² + 3x  =   x • (x + 3) 

Adding fractions that have a common denominator :

 3.2       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+3) • x2 - (28)     x4 + 3x3 - 28
 —————————————————————  =  —————————————
          x2                    x2      

Equation at the end of step  3  :

         7          (x4+3x3-28)
  (((x+————)+3x)-10)———————————-25)
       (x2)(            x2     

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

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

          25     25 • x2
    25 =  ——  =  ———————
          1        x2   

Polynomial Roots Calculator :

 4.2    Find roots (zeroes) of :       F(x) = x⁴ + 3x³ - 28
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  -28.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,7 ,14 ,28

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 4.3       Adding up the two equivalent fractions

 (x4+3x3-28) - (25 • x2)     x4 + 3x3 - 25x2 - 28
 ———————————————————————  =  ————————————————————
           x2                         x2         

Equation at the end of step  4  :

         7          (x4+3x3-25x2-28)
  (((x+————)+3x)-10)————————————————
       (x2)                x2       
 Step  5  :
             7
 Simplify   ——
            x2

Equation at the end of step  5  :

   7                (x4 + 3x3 - 25x2 - 28)
  ——) + 3x) - 10) ÷ ——————————————————————
  x2                          x2          

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a fraction to a whole

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

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

Adding fractions that have a common denominator :

 6.2       Adding up the two equivalent fractions

 x • x2 + 7     x3 + 7
 ——————————  =  ——————
     x2           x2  

Equation at the end of step  6  :

  (x3 + 7)               (x4 + 3x3 - 25x2 - 28)
  ———————— + 3x) - 10) ÷ ——————————————————————
     x2                            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 :

          3x     3x • x2
    3x =  ——  =  ———————
          1        x2   

Trying to factor as a Sum of Cubes :

 7.2      Factoring:  x³ + 7 

Theory : A sum 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 :

 7.3    Find roots (zeroes) of :       F(x) = x³ + 7

     See theory in step 4.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  7.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 7.4       Adding up the two equivalent fractions

 (x3+7) + 3x • x2     4x3 + 7
 ————————————————  =  ———————
        x2              x2   

Equation at the end of step  7  :

  (4x3 + 7)         (x4 + 3x3 - 25x2 - 28)
  ————————— - 10) ÷ ——————————————————————
     x2                       x2          

Step  8  :

Rewriting the whole as an Equivalent Fraction :

 8.1   Subtracting a whole from a fraction

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

          10     10 • x2
    10 =  ——  =  ———————
          1        x2   

Trying to factor as a Sum of Cubes :

 8.2      Factoring:  4x³ + 7 

Check :  4  is not a cube !!

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

Polynomial Roots Calculator :

 8.3    Find roots (zeroes) of :       F(x) = 4x³ + 7

     See theory in step 4.2
In this case, the Leading Coefficient is  4  and the Trailing Constant is  7.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1 ,7

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 8.4       Adding up the two equivalent fractions

 (4x3+7) - (10 • x2)     4x3 - 10x2 + 7
 ———————————————————  =  ——————————————
         x2                    x2      

Equation at the end of step  8  :

  (4x3 - 10x2 + 7)   (x4 + 3x3 - 25x2 - 28)
  ———————————————— ÷ ——————————————————————
         x2                    x2          

Step  9  :

         4x3-10x2+7      x4+3x3-25x2-28
 Divide  ——————————  by  ——————————————
             x2                x2      

 9.1    Dividing fractions

To divide fractions, write the divison as multiplication by the reciprocal of the divisor :

4x3 - 10x2 + 7     x4 + 3x3 - 25x2 - 28       4x3 - 10x2 + 7               x2          
——————————————  ÷  ————————————————————   =   ——————————————  •  ——————————————————————
      x2                    x2                      x2           (x4 + 3x3 - 25x2 - 28)

Checking for a perfect cube :

 9.2    x⁴ + 3x³ - 25x² - 28  is not a perfect cube

Trying to factor by pulling out :

 9.3      Factoring:  x⁴ + 3x³ - 25x² - 28 

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

Group 1:  -25x² - 28 
Group 2:  x⁴ + 3x³ 

Pull out from each group separately :

Group 1:   (25x² + 28) • (-1)
Group 2:   (x + 3) • (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 :

 9.4    Find roots (zeroes) of :       F(x) = 4x³ - 10x² + 7

     See theory in step 4.2
In this case, the Leading Coefficient is  4  and the Trailing Constant is  7.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1 ,7

 Let us test ....

Polynomial Roots Calculator found no rational roots

Polynomial Roots Calculator :

 9.5    Find roots (zeroes) of :       F(x) = x⁴ + 3x³ - 25x² - 28

     See theory in step 4.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -28.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,7 ,14 ,28

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

     4x3 - 10x2 + 7   
  ————————————————————
  x4 + 3x3 - 25x2 - 28