Solution - Finding the roots of polynomials

Step  1  :

            64
 Simplify   ——
            x 

Equation at the end of step  1  :

                    64
  (((x4)-(12•(x2)))-——)+4
                    x 
 Step  2  :

Equation at the end of step  2  :

                         64     
  (((x4) -  (22•3x2)) -  ——) +  4
                         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  x  as the denominator :

                  x4 - 12x2     (x4 - 12x2) • x
     x4 - 12x2 =  —————————  =  ———————————————
                      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

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   x⁴ - 12x²  =   x² • (x² - 12) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  x² - 12 

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 : 12 is not a square !!

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

Adding fractions that have a common denominator :

 4.3       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:

 x2 • (x2-12) • x - (64)     x5 - 12x3 - 64
 ———————————————————————  =  ——————————————
            x                      x       

Equation at the end of step  4  :

  (x5 - 12x3 - 64)    
  ———————————————— +  4
         x            

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 :

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

Polynomial Roots Calculator :

 5.2    Find roots (zeroes) of :       F(x) = x⁵ - 12x³ - 64
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  -64.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8 ,16 ,32 ,64

 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
   x⁵ - 12x³ - 64 
can be divided with  x + 2 

Polynomial Long Division :

 5.3    Polynomial Long Division
Dividing :  x⁵ - 12x³ - 64 
                              ("Dividend")
By         :    x + 2    ("Divisor")

Quotient :  x⁴-2x³-8x²+16x-32  Remainder:  0 

Polynomial Roots Calculator :

 5.4    Find roots (zeroes) of :       F(x) = x⁴-2x³-8x²+16x-32

     See theory in step 5.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -32.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8 ,16 ,32

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 5.5       Adding up the two equivalent fractions

 (x4-2x3-8x2+16x-32) • (x+2) + 4 • x      x5 - 12x3 + 4x - 64
 ———————————————————————————————————  =  ———————————————————
                  x                               x         

Checking for a perfect cube :

 5.6    x⁵ - 12x³ + 4x - 64  is not a perfect cube

Trying to factor by pulling out :

 5.7      Factoring:  x⁵ - 12x³ + 4x - 64 

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

Group 1:  4x - 64 
Group 2:  -12x³ + x⁵ 

Pull out from each group separately :

Group 1:   (x - 16) • (4)
Group 2:   (x² - 12) • (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 :

 5.8    Find roots (zeroes) of :       F(x) = x⁵ - 12x³ + 4x - 64

     See theory in step 5.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -64.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8 ,16 ,32 ,64

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  x5  12x3 + 4x - 64
  ———————————————————
           x