Solution - Adding, subtracting and finding the least common multiple

Step  1  :

            x2
 Simplify   ——
            x2

Canceling Out :

 1.1    Canceling out x² as it appears on both sides of the fraction line

Equation at the end of step  1  :

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

Equation at the end of step  2  :

            x           4
  ((((((2•————)-4)+2x)+——)-4)+1)-4
          (x2)         x2
 Step  3  :
             x
 Simplify   ——
            x2

Dividing exponential expressions :

 3.1    x¹ divided by x² = x^{(1 - 2)} = x^(-1) = ¹/_x¹ = ¹/_x

Equation at the end of step  3  :

          1          4
  ((((((2•—)-4)+2x)+——)-4)+1)-4
          x         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  x  as the denominator :

         4     4 • x
    4 =  —  =  —————
         1       x  

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

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:

 2 - (4 • x)     2 - 4x
 ———————————  =  ——————
      x            x   

Equation at the end of step  4  :

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

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Adding a whole to a fraction

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

          2x     2x • x
    2x =  ——  =  ——————
          1        x   

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   2 - 4x  =   -2 • (2x - 1) 

Adding fractions that have a common denominator :

 6.2       Adding up the two equivalent fractions

 -2 • (2x-1) + 2x • x     2x2 - 4x + 2
 ————————————————————  =  ————————————
          x                    x      

Equation at the end of step  6  :

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

Step  7  :

Step  8  :

Pulling out like terms :

 8.1     Pull out like factors :

   2x² - 4x + 2  =   2 • (x² - 2x + 1) 

Trying to factor by splitting the middle term

 8.2     Factoring  x² - 2x + 1 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -2x  its coefficient is  -2 .
The last term, "the constant", is  +1 

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

Step-2 : Find two factors of  1  whose sum equals the coefficient of the middle term, which is   -2 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -1  and  -1 
                     x² - 1x - 1x - 1

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-1)
              Add up the last 2 terms, pulling out common factors :
                     1 • (x-1)
Step-5 : Add up the four terms of step 4 :
                    (x-1)  •  (x-1)
             Which is the desired factorization

Multiplying Exponential Expressions :

 8.3    Multiply  (x-1)  by  (x-1) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (x-1)  and the exponents are :
          1 , as  (x-1)  is the same number as  (x-1)¹ 
 and   1 , as  (x-1)  is the same number as  (x-1)¹ 
The product is therefore,  (x-1)⁽¹⁺¹⁾ = (x-1)² 

Calculating the Least Common Multiple :

 8.4    Find the Least Common Multiple

      The left denominator is :       x 

      The right denominator is :       x² 

      Least Common Multiple:
      x² 

Calculating Multipliers :

 8.5    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 = x

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 8.6      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.      2 • (x-1)2 • x
   ——————————————————  =   ——————————————
         L.C.M                   x2      

   R. Mult. • R. Num.       4
   ——————————————————  =   ——
         L.C.M             x2

Adding fractions that have a common denominator :

 8.7       Adding up the two equivalent fractions

 2 • (x-1)2 • x + 4     2x3 - 4x2 + 2x + 4
 ——————————————————  =  ——————————————————
         x2                     x2        

Equation at the end of step  8  :

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

Step  9  :

Rewriting the whole as an Equivalent Fraction :

 9.1   Subtracting a whole from a fraction

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

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

Step  10  :

Pulling out like terms :

 10.1     Pull out like factors :

   2x³ - 4x² + 2x + 4  = 

  2 • (x³ - 2x² + x + 2) 

Checking for a perfect cube :

 10.2    x³ - 2x² + x + 2  is not a perfect cube

Trying to factor by pulling out :

 10.3      Factoring:  x³ - 2x² + x + 2 

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

Group 1:  x + 2 
Group 2:  x³ - 2x² 

Pull out from each group separately :

Group 1:   (x + 2) • (1)
Group 2:   (x - 2) • (x²)

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 :

 10.4    Find roots (zeroes) of :       F(x) = x³ - 2x² + x + 2
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  2.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 10.5       Adding up the two equivalent fractions

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

Equation at the end of step  10  :

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

Step  11  :

Rewriting the whole as an Equivalent Fraction :

 11.1   Adding a whole to a fraction

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

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

Step  12  :

Pulling out like terms :

 12.1     Pull out like factors :

   2x³ - 8x² + 2x + 4  = 

  2 • (x³ - 4x² + x + 2) 

Checking for a perfect cube :

 12.2    x³ - 4x² + x + 2  is not a perfect cube

Trying to factor by pulling out :

 12.3      Factoring:  x³ - 4x² + x + 2 

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

Group 1:  x + 2 
Group 2:  x³ - 4x² 

Pull out from each group separately :

Group 1:   (x + 2) • (1)
Group 2:   (x - 4) • (x²)

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 :

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

     See theory in step 10.4
In this case, the Leading Coefficient is  1  and the Trailing Constant is  2.

 The factor(s) are:

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

 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
   x³ - 4x² + x + 2 
can be divided with  x - 1 

Polynomial Long Division :

 12.5    Polynomial Long Division
Dividing :  x³ - 4x² + x + 2 
                              ("Dividend")
By         :    x - 1    ("Divisor")

Quotient :  x²-3x-2  Remainder:  0 

Trying to factor by splitting the middle term

 12.6     Factoring  x²-3x-2 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -3x  its coefficient is  -3 .
The last term, "the constant", is  -2 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -2 = -2 

Step-2 : Find two factors of  -2  whose sum equals the coefficient of the middle term, which is   -3 .

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

Adding fractions that have a common denominator :

 12.7       Adding up the two equivalent fractions

 2 • (x2-3x-2) • (x-1) + x2     2x3 - 7x2 + 2x + 4
 ——————————————————————————  =  ——————————————————
             x2                         x2        

Equation at the end of step  12  :

  (2x3 - 7x2 + 2x + 4)    
  ———————————————————— -  4
           x2             

Step  13  :

Rewriting the whole as an Equivalent Fraction :

 13.1   Subtracting a whole from a fraction

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

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

Checking for a perfect cube :

 13.2    2x³ - 7x² + 2x + 4  is not a perfect cube

Trying to factor by pulling out :

 13.3      Factoring:  2x³ - 7x² + 2x + 4 

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

Group 1:  2x + 4 
Group 2:  2x³ - 7x² 

Pull out from each group separately :

Group 1:   (x + 2) • (2)
Group 2:   (2x - 7) • (x²)

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 :

 13.4    Find roots (zeroes) of :       F(x) = 2x³ - 7x² + 2x + 4

     See theory in step 10.4
In this case, the Leading Coefficient is  2  and the Trailing Constant is  4.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 13.5       Adding up the two equivalent fractions

 (2x3-7x2+2x+4) - (4 • x2)     2x3 - 11x2 + 2x + 4
 —————————————————————————  =  ———————————————————
            x2                         x2         

Checking for a perfect cube :

 13.6    2x³ - 11x² + 2x + 4  is not a perfect cube

Trying to factor by pulling out :

 13.7      Factoring:  2x³ - 11x² + 2x + 4 

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

Group 1:  2x + 4 
Group 2:  2x³ - 11x² 

Pull out from each group separately :

Group 1:   (x + 2) • (2)
Group 2:   (2x - 11) • (x²)

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 :

 13.8    Find roots (zeroes) of :       F(x) = 2x³ - 11x² + 2x + 4

     See theory in step 10.4
In this case, the Leading Coefficient is  2  and the Trailing Constant is  4.

 The factor(s) are:

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

 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
   2x³ - 11x² + 2x + 4 
can be divided with  2x + 1 

Polynomial Long Division :

 13.9    Polynomial Long Division
Dividing :  2x³ - 11x² + 2x + 4 
                              ("Dividend")
By         :    2x + 1    ("Divisor")

Quotient :  x²-6x+4  Remainder:  0 

Trying to factor by splitting the middle term

 13.10     Factoring  x²-6x+4 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -6x  its coefficient is  -6 .
The last term, "the constant", is  +4 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 4 = 4 

Step-2 : Find two factors of  4  whose sum equals the coefficient of the middle term, which is   -6 .

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

Final result :

  (x2 - 6x + 4) • (2x + 1)
  ————————————————————————
             x2